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+{"text":"\\section{Introduction}\n\\IEEEPARstart{D}{igital} pulse interval modulation (DPIM) \\cite{Ghassemlooy2012} is an energy-efficient modulation technique for optical wireless communications based on intensity modulation and direct detection (IM\/DD). Compared with conventional on-off keying (OOK), pulse amplitude modulation (PAM), pulse width modulation (PWM) and pulse position modulation (PPM), it utilizes pulse intervals to carry data and does not require accurate symbol-level synchronization.\nSince proposed, it has been widely investigated in spectral evaluation \\cite{Cariolaro2001,Ono2001} and error performance over various channels \\cite{Zhang2000,Ghassemlooy2001,Ghassemlooy2003,Ghassemlooy2005,Popoola2006,Ghassemlooy2008,Ma2010,Jin2010,Jiang2014,Liu2014,Xu2015,Dabiri2016,Das2019}. Many enhanced modulation schemes have been proposed\n\\cite{Ghassemlooy2006,Aldibbiat2001,Ono2002,Ono2004,Sethakaset2005,Numata2011,He2013,Abdullah2014,Liao2014,Mi2016}. For instance, the authors of \\cite{Ghassemlooy2006}  and \\cite{Aldibbiat2001} proposed multilevel digital pulse interval modulation (MDPIM) and dual-header pulse interval modulation (DHPIM), which improved the information-carrying capability by additionally varying  pulse amplitudes and  pulse widths, respectively.\n\n\n\\begin{figure*}[t]\n  \\centering\n \n    \\includegraphics[width=0.9\\textwidth]{system_model_new}\\\\\n  \\caption{System model of DPIM}\n  \\label{System_Model}\n\\end{figure*}\n\nDPIM symbols do not have equal duration. The fact complicates the analysis and application of DPIM.  Note that a single-chip error not only corrupts the bits directly associated with that chip but also shifts the bits that follow those bits \\cite{Shiu1999}. Another distinct property resulting from the nonuniform symbol length is that the symbol boundaries are\nnot known before detection. Conventional soft decoding approaches for invariant symbol duration such as the Viterbi algorithm are not applicable \\cite{Shiu1999}. The optimal soft decoding of\nDPIM requires the use of high-complexity maximum likelihood sequence detector (MLSD). Therefore, most practical implementations of\nDPIM would probably employ hard-decision decoding, i.e., sample-by-sample optimal threshold detection (OTD). Even though OTD has low complexity, it suffers from more severe bit error propagation, which leads to high bit error ratio (BER) in an erroneous packet. When using forward error correction (FEC) codes to correct the erroneous packet, lots of redundancy must be added in codewords because of the high BER thus resulting in a low rate. In other words, because of the high BER in an erroneous packet, DPIM can not benefit from the association with high-rate FEC codes for packet correction. To fix this problem, the error propagation of DPIM should be mitigated. \nMotivated by improving the system reliability, this paper proposes a low-complexity detection method that achieves the same performance as MLSD leveraging the sparsity of  DPIM sequences. Moreover, the paper proposes a barrier signal design for DPIM to limit the error propagation. It should be mentioned that\nwe adopt the average BER metric to measure the system reliability in this paper, which is defined by the packet error rate (PER) multiplying the expected BER in an erroneous packet.  Compared with the PER metric used in  \\cite{Zhang2000,Ghassemlooy2001,Ghassemlooy2003,Ghassemlooy2005,Popoola2006,Ghassemlooy2008,Ma2010,Jin2010,Jiang2014,Liu2014,Xu2015,Dabiri2016,Das2019},\nthe average BER metric can measure both the PER and the error propagation level in an erroneous packet. The contributions of this paper are summarized as follows.\n \\begin{itemize}\n\\item At the receiver side, the optimal MLSD of DPIM is formulated as a sparsity-constrained least square minimization problem. An ordered sequence detection (OSD) is proposed to solve the problem with low complexity.  The computational complexity and detection delay of OSD are analyzed. The relationship between OSD and the orthogonal matching pursuit (OMP) algorithm in \\cite{Tropp2007} is revealed, which explains why OSD can achieve the same performance as MLSD.\n\\item At the transmitter side, a barrier signal-aided digital pulse interval modulation (BDPIM) is proposed to limit the error propagation and jointly utilized with OSD.  For large packet detection, we propose to use a combination of OTD and OSD to reduce the storage delay.  The parameter optimization of BDPIM is investigated by simulations.  \n\n\\item Approximate upper bounds of the average BER performance of DPIM with OTD (DPIM-OTD), DPIM with OSD (DPIM-OSD), BDPIM with OSD (BDPIM-OSD) and BDPIM with a combination of OTD and OSD (BDPIM-OTD-OSD) are derived.  Compared with \\cite{Guo2018d}, the approximate BER upper bounds provided in this paper are applicable to all SNR regimes. Simulations are conducted to corroborate our theoretical analysis.\n\n\\item Comparisons are made among coded and uncoded  DPIM-OTD, DPIM-OSD, BDPIM-OSD,  BDPIM-OTD-OSD, etc. For extension, we investigate the performance evaluation over Gamma-Gamma turbulence channels. \n \\end{itemize} \n\n \n The remainder of the paper is organized as follows.  Section II describes the system model and the proposed designs. Section III presents the approximate upper bounds of the average BER performance of all described systems.\nIn Section IV, we show the superiority of the proposed schemes by simulations, investigate the impact of parameter settings on performance and discuss the extension to Gamma-Gamma  turbulence channels. Conclusions are drawn in the last section.\n\nIn this paper, the normal letter $x$ represents a scalar. The boldface letter $\\textbf{x}$ stands for a vector. $||\\textbf{x}||_0$ and $||\\textbf{x}||_2$ denote the $l_0$ norm and $l_2$ norm of vector $\\textbf{x}$, respectively. $\\mathbb{R}$ represents the real-number field. $\\mathbb{E}(\\cdot)$ represents the expectation operation. $\\lfloor x\\rceil$ denotes the nearest integer to $x$. $\\ln$ and $\\log$ stand for the logarithm functions of base $e$ and base $2$, respectively. $\\mathcal{Q}(\\cdot)$ denotes the tail distribution function of the standard normal distribution. $\\mathrm{erf}(\\cdot)$ is the Gaussian error function. $n~\\mathrm{mod}~K=0$ when $n$ is a multiple of $K$ and otherwise $n~\\mathrm{mod}~K\\neq0$. \n\n \n\n\\begin{figure}[t]\n  \\centering\n \n \\includegraphics[width=0.42\\textwidth]{signal_form_new}\\\\\n  \\caption{The signal forms of DPIM}\n  \\label{Signal_form}\n  \\end{figure}\n  \n\\section{System Model and Proposed Designs}\nIn this section, the system model of conventional DPIM is first presented and then followed by an introduction to existing detection methods (i.e., MLSD, OTD) and the proposed OSD.  Also,   the new signal design, i.e., BDPIM is  described. \n\\subsection{System Model of DPIM}\nWe consider a DPIM system as illustrated in Fig. \\ref{System_Model}, where  $N_s$-symbol packets are transmitted by using $M$-ary DPIM.  The signal forms for DPIM has two types with and without guard intervals (GIs).\nTo demonstrate their differences,  we illustrate the signal forms of $4$-DPIM  in Fig. \\ref{Signal_form}.  The data bits $\\{00,~01,~10,~11\\}$ are mapped to  $\\{\\mathrm{A},~ \\mathrm{A}0,~\\mathrm{A}00,~\\mathrm{A}000\\}$ with no GI, and to $\\{\\mathrm{A}0,~\\mathrm{A}00,~\\mathrm{A}000,~\\mathrm{A}0000\\}$ with $1$ GI, where $\\mathrm{A}$ represents the peak amplitude and is proportional to the optical power. For the sake of simplicity, we assume that the proportional coefficient is $1$. In this paper, a general case with $g$ ($g\\in\\mathbb{N}$) GIs is considered\nand the average symbol duration is given by\n\\begin{equation}\nL_s=\\frac{1}{M}\\sum_{l=1}^{M}l+g=\\frac{M+2g+1}{2}.\n\\end{equation}\nNote that because the symbol duration of DPIM is variant, the total sequence length denoted by $L$ is also variant and the expected sequence length of $N_s$-symbol packets is $\\mathbb{E}\\{L\\}=N_sL_s$.\n\n Let $h$ denote the channel coefficient.  The received signal vector $\\textbf{y}\\in\\mathbb{R}^{L}$ can be written as\n\\begin{equation}\\label{sig_model}\n\\textbf{y}=h\\textbf{x}+\\textbf{n},\n\\end{equation}\nwhere $\\textbf{x}\\in\\mathbb{R}^{L}$ represents the transmit signal vector, and $\\textbf{n}\\in\\mathbb{R}^{L}$ is real-valued additive white Gaussian noise (AWGN) with zero mean and variance $\\sigma_n^2\\textbf{I}_L$. Moreover, the electrical SNR is denoted as $\\gamma$ which is expressed as $\\gamma=\\mathrm{A}^2\/\\sigma_n^2$.\n\\subsection{Detection Methods}\n\\subsubsection{MLSD}\nIt is assumed that $h$ is known at the receiver. With full knowledge of $h$, $N_s$ and $L$, MLSD is given by\n\\begin{equation}\n\\begin{split}\n(\\textbf{P1}):\\hat{\\textbf{x}}&=\\arg\\max_{||\\textbf{x}||_0=N_s}p_{\\textbf{Y}}(\\textbf{y}|\\textbf{x},h)\\\\\n&=\\arg\\min_{||\\textbf{x}||_0=N_s}||\\textbf{y}-h\\textbf{x}||_2^2,\n\\end{split}\n\\end{equation}\nsince the probability density function (PDF) of \\textbf{y} conditioned\non \\textbf{x} and $h$ is\n\\begin{equation}\np_{\\textbf{Y}}(\\textbf{y}|\\textbf{x},h) \\propto \\exp(-||\\textbf{y}-h\\textbf{x}||_2^2).\n\\end{equation}\nThere are a total of $\\left(L\\atop N_s\\right)$ feasible solutions for  $\\hat{\\textbf{x}}$. For each feasible solution, $L$ real-number multiplications are needed to compute the $l_2$ norm. Therefore, MLSD based on exhaustive search requires $\\left(L\\atop N_s\\right)L$ multiplications. Due to the high computational complexity,  it  is prohibitive especially for large packet detection.\n\n\\subsubsection{OTD} OTD detects the sequence using an optimal threshold $h\\mathrm{A}_\\mathrm{T}$. The key of OTD is finding $\\mathrm{A}_\\mathrm{T}$ based on the minimizing chip-error probability criterion. Specifically, the chip error probability can be expressed as\n\\begin{equation}\n\\begin{split}\nP_c=&\\mathrm{Pr}(0)\\mathrm{Pr}(0\\rightarrow \\mathrm{A})+\\mathrm{Pr}(\\mathrm{A})\\mathrm{Pr}(\\mathrm{A} \\rightarrow 0)\\\\\n=&\\mathrm{Pr}(0)\\int_{\\mathrm{A}_\\mathrm{T}}^{\\infty}\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2x^2}{2\\sigma_n^2}}dx\n\\\\&~+\\mathrm{Pr}(\\mathrm{A})\\int^{\\mathrm{A}_\\mathrm{T}}_{-\\infty}\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2(x-\\mathrm{A})^2}{2\\sigma_n^2}}dx.\n\\end{split}\n\\end{equation}\nTaking the first derivative of $P_c$ with respect to $A_{T}$, we have\n\\begin{equation}\n\\frac{\\partial P_c}{\\partial\\mathrm{A}_\\mathrm{T}}=\\mathrm{Pr}(\\mathrm{A})\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2(\\mathrm{A}-\\mathrm{A}_{\\mathrm{T}})^2}{2\\sigma_n^2}} -\\mathrm{Pr}(0)\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2\\mathrm{A}_{\\mathrm{T}}^2}{2\\sigma_n^2}}.\n\\end{equation}\nBy solving the equation $\\frac{\\partial P_c}{\\partial\\mathrm{A}_\\mathrm{T}}=0$, \nwe obtain the optimal threshold $\\mathrm{A}_{\\mathrm{T}}$ as\n\\begin{equation}\n\\mathrm{A}_{\\mathrm{T}}=\\frac{\\mathrm{A}}{2}-\\frac{\\mathrm{A}}{h^2\\gamma}\\ln\\frac{\\mathrm{Pr}(\\mathrm{A})}{\\mathrm{Pr}(0)}.\n\\end{equation}\nFrom  (\\ref{eqAt}), it is observed that $\\mathrm{A}_{\\mathrm{T}}$ is determined by $\\mathrm{Pr}(\\mathrm{A})$ and $\\mathrm{Pr}(\\mathrm{0})$, which are expressed as\n\\begin{equation}\\label{PrA}\n\\mathrm{Pr}(\\mathrm{A})=\\frac{N_s}{L},\n\\end{equation}\nand\n\\begin{equation}\\label{Pr0}\n\\mathrm{Pr}(0)=\\frac{L-N_s}{L},\n\\end{equation}\nrespectively. From (\\ref{PrA}) and (\\ref{Pr0}), it is observed that $N_s$ and $L$ should be be known for the determination of $\\mathrm{A}_\\mathrm{T}$. In practical implementation, the exact $\\mathrm{Pr}(\\mathrm{A})$, $\\mathrm{Pr}(0)$ can be approximately given by their expectations which are $\\mathbb{E}\\{\\mathrm{Pr}(\\mathrm{A})\\}=1\/L_s$ and $\\mathbb{E}\\{\\mathrm{Pr}(\\mathrm{A})\\}=1-1\/L_s$. Based on these approximations, $\\mathrm{A}_{\\mathrm{T}}$ can be determined without the exact knowledge of $N_s$ and $L$ as\n\\begin{equation}\\label{eqAt}\n\\mathrm{A}_{\\mathrm{T}}\\approx\\frac{\\mathrm{A}}{2}+\\frac{ \\mathrm{A}}{h^2\\gamma}\\ln{(L_s-1)}.\n\\end{equation}\n\\subsubsection{OSD} The optimal MLSD expressed as (\\textbf{P1}) is indeed a sparsity-constrained least square minimization problem and the average sparsity level of DPIM sequences is $\\mathbb{E}\\{N_s\/L\\}=1\/L_s$. According to \\cite{Tropp2007}, the problem can be well solved by the OMP algorithm.\nIn use of the OMP algorithm,  the sequence can be iteratively detected as\n\\begin{equation}\n\\hat{t}_k=\\arg \\max_{t=1,2,\\cdots,L} |\\langle\\textbf{r}_{k},\\textbf{e}_t\\rangle|\n\\end{equation}\nwith $\\textbf{r}_1=\\textbf{y}$ and  $\\textbf{r}_{k}$ being updated by\n\\begin{equation}\n\\textbf{r}_{k+1}=\\textbf{r}_{k}-\\langle\\textbf{r}_{k},\\textbf{e}_{\\hat{t}_k}\\rangle \\textbf{e}_{\\hat{t}_k},\n\\end{equation}\nwhere $\\hat{t}_k$ is the position of $\\mathrm{A}$ detected in the $k$th iteration, $\\textbf{e}_t\\in\\mathbb{R}^L$ denotes the $t$th basis vector with $t$th entry being one and others being zeros.  It is observed from the procedure that OMP detects the $k$th largest receive signal as $\\mathrm{A}$ in the $k$th iteration and repeats the detection until the number of iterations  equals to $N_s$.\n Instead of using the iterative procedure, we propose a simplified equivalent OSD in this paper. In OSD, a sorting algorithm is used to sort $\\mathbf{y}=[y(1),y(2),\\cdots,y(L)]$, the $N_s$ largest of which are detected as $\\mathrm{A}$ and the others as $0$. Let $\\{i(t)\\}$ represent the order of $\\{y(t)\\}$, where $i(t)=1$ means that $y(t)$ is the largest. Mathematically,  the detected signal sequence can be written as\n\\begin{equation}\nd(t)=\\begin{cases}\n\\mathrm{A}, & 1\\leq i(t) \\leq N_s,\\\\\n0, &N_s+1\\leq i(t) \\leq L.\n\\end{cases}\n\\end{equation}\nThen by inputting $\\mathbf{d}=[d(1),d(2),\\cdots,d(L)]$ into conventional DPIM demodulator, we finally obtain the data bits. Moreover, it should be mentioned that $h$ is not used in the OSD procedure, which means that channel estimation is not required.\n\\begin{table*}[t]\n\\centering\n\\caption{Comparisons of Three Detection Methods}\\label{tab1}\n\\footnotesize\n    \\begin{tabular}{ | c | c | c | c|c|c|}\n    \\hline\n    Method & Detection manner & \\tabincell{c}{Computation \\\\ complexity\/delay} & Storage delay  &Performance & \\tabincell{c}{Required information} \\\\ \\hline\n    MLSD & packet-by-packet &  $O\\left(\\left(L\\atop N_s\\right)L\\right)$,  High & Low if $L$ is small &Optimal&$h$, $L$, $N_s$\\\\ \\hline\n   OTD & sample-by-sample &  $O\\left(L\\right)$, Low & Low&Suboptimal&$h$, $L_s$\\\\ \\hline\n   OSD & packet-by-packet &  $O\\left(L\\log L\\right)$, Low  & Low if $L$ is small&Optimal&$L$, $N_s$\\\\ \\hline\n    \\end{tabular}\n\\end{table*}\n\\begin{table*}[]\n\\caption{Data mapping of PPM, DPIM, MDPIM, DHPIM and BDPIM}\n\\center\n\\begin{tabular}{lllllll}\nData& PPM & DPIM & MDPIM\\cite{Ghassemlooy2006} & $\\textrm{DHPIM}$\\cite{Aldibbiat2001}&  $\\textrm{BDPIM}_{n~\\mathrm{mod}~K\\neq0}$&$\\textrm{BDPIM}_{n~\\mathrm{mod}~K=0}$\\\\\n 000&$\\mathrm{A}$0000000&$\\mathrm{A}$0&$\\mathrm{A}_{\\mathrm{L}}0$ &$\\mathrm{A}$0& $\\mathrm{A}_{\\mathrm{L}}$0&$\\mathrm{A}_{\\mathrm{H}}$0\\\\\n 001&0$\\mathrm{A}$000000&$\\mathrm{A}$00 &$\\mathrm{A}_{\\mathrm{L}}$00  &$\\mathrm{A}$00 &$\\mathrm{A}_{\\mathrm{L}}$00&$\\mathrm{A}_{\\mathrm{H}}$00\\\\\n 010&00$\\mathrm{A}$00000&$\\mathrm{A}$000&$\\mathrm{A}_{\\mathrm{L}}$000& $\\mathrm{A}$000&$\\mathrm{A}_{\\mathrm{L}}$000&$\\mathrm{A}_{\\mathrm{H}}$000\\\\\n 011&000$\\mathrm{A}$0000&$\\mathrm{A}$0000&$\\mathrm{A}_{\\mathrm{L}}$0000&$\\mathrm{A}$0000&$\\mathrm{A}_{\\mathrm{L}}$0000&$\\mathrm{A}_{\\mathrm{H}}$0000\\\\\n100&0000$\\mathrm{A}$000&$\\mathrm{A}$00000&$\\mathrm{A}_{\\mathrm{H}}$0&$\\mathrm{A}$$\\mathrm{A}$0& $\\mathrm{A}_{\\mathrm{L}}$00000&$\\mathrm{A}_{\\mathrm{H}}$00000\\\\\n 101&00000$\\mathrm{A}$00&$\\mathrm{A}$000000 &$\\mathrm{A}_{\\mathrm{H}}$00  &$\\mathrm{A}$$\\mathrm{A}$00 &$\\mathrm{A}_{\\mathrm{L}}$000000&$\\mathrm{A}_{\\mathrm{H}}$000000\\\\\n 110&000000$\\mathrm{A}$0&$\\mathrm{A}$0000000& $\\mathrm{A}_{\\mathrm{H}}$000 &$\\mathrm{A}$$\\mathrm{A}$000 &$\\mathrm{A}_{\\mathrm{L}}$0000000&$\\mathrm{A}_{\\mathrm{H}}$0000000\\\\\n 111&0000000$\\mathrm{A}$&$\\mathrm{A}$00000000&$\\mathrm{A}_{\\mathrm{H}}$0000&$\\mathrm{A}$$\\mathrm{A}$0000  &$\\mathrm{A}_{\\mathrm{L}}$00000000&$\\mathrm{A}_{\\mathrm{H}}$00000000\\\\\n\\end{tabular}\n\\end{table*}\n\n\\emph{Computational Complexity and Delay Analysis of Three Detection Methods:} As analyzed in Section II-B-1,  MLSD based on exhaustive search requires $\\left(L\\atop N_s\\right)L$ multiplications, which can be expressed as $\\mathcal{O}\\left(\\left(L\\atop N_s\\right)L\\right)$.\nOTD requires $L$ comparisons, the complexity of which can be given by $\\mathcal{O}\\left(L\\right)$. OSD can sort $L$ received signals by using the QuickSort algorithm, whose complexity is $\\mathcal{O}(L\\log L)$.  \nThe detection delay includes the storage latency and the computation delay. Since MLSD and OSD are operated packet-by-packet, the storage delay of an $L$-chip packet is\n\\begin{equation}\\label{tau1}\n\\tau_{1}=\\frac{L}{R_{c}},\n\\end{equation}\nwhere $R_{c}$ is the transmission rate in chips per second. Differently, OTD is performed sample-by-sample, and thus the storage delay of OTD is\n\\begin{equation}\n\\tau_{2}=\\frac{1}{R_{c}}.\n\\end{equation}\nFor high-rate optical wireless communications, the storage latency of OTD only depends on the transmission rate and thus it is low regardless of the sequence length. The latency of OSD and that of MLSD are also dependent on $L$, and therefore they are low only if $L$ is small.  The computational delay is proportional to the computational complexity. Since the complexity of OTD and OSD is quite low, the computation delay can be omitted. On the contrary, the complexity of MLSD is high thus resulting in a computation delay of $O\\left(\\left(L\\atop N_s\\right)L\\right)$ unit times (UTs), where a UT is the amount of time it takes for a multiplication. For clearly viewing the differences of these detection methods, we list the comparison results in terms of complexity, storage delay, performance as well as the required information in Table I.\n\n\n\n\\subsection{Barrier Signal Design}\nThe severe error propagation of DPIM mainly results from unknown symbol boundaries. To overcome it, we propose a barrier signal design, i.e., BDPIM.  As shown in  Fig. \\ref{Packet}, BDPIM signals have two amplitudes. The amplitude of the $n$th symbol is $\\mathrm{A}_{\\mathrm{H}}$, when $n~\\mathrm{mod}~K=0$  and otherwise $\\mathrm{A}_{\\mathrm{L}}$ in BDPIM, where $\\mathrm{A}_{\\mathrm{H}}>\\mathrm{A}_{\\mathrm{L}}$ and $K$ is a positive integer.  For fair comparison with DPIM under the same optical power constraint, the power allocation in every $K$ symbols of BDPIM  satisfies \n\\begin{equation}\\label{Con1}\n(K-1)\\mathrm{A}_{\\mathrm{L}}+\\mathrm{A}_{\\mathrm{H}}=K\\mathrm{A}.\n\\end{equation}\n To show the difference of the proposed design from existing signal designs, we list the data mapping of PPM, DPIM with 1 GI, MDPIM \\cite{Ghassemlooy2006}, DHPIM\\cite{Aldibbiat2001}, the proposed BDPIM in Table II. It is shown that BDPIM  has the same average symbol duration as DPIM, which is larger than that of MDPIM \\cite{Ghassemlooy2006} and that of DHPIM \\cite{Aldibbiat2001}.\nFor simplicity, we assume that the number of symbols $N_s$ in a packet can be divided by $K$, which means $N_s$ can be expressed as $N_s=KQ$ and $Q$ is a positive integer.\n\\begin{figure}[t]\n  \\centering\n \n    \\includegraphics[width=0.4\\textwidth]{frame_structure_new}\\\\\n  \\caption{The signal forms of BDPIM}\n  \\label{Packet}\n\\end{figure}\n\n\nAt the receiver, OSD for BDPIM has two phases. In the first OSD phase, $\\mathbf{y}$ is sorted and the $Q$ largest are detected as $\\mathrm{A}_{\\mathrm{H}}$, the positions of which is recorded as $t_1, t_2,\\cdots, t_Q$.  Let $t_1'< t_2' <\\cdots< t_Q'$ denote the sorted positions in an ascent order and $t'_0=0$. In the second OSD phase,  $\\mathbf{y}(t_i'+1:t_{i+1}'-1)$ is sorted, the $(K-1)$ largest values of which are detected as $\\mathrm{A}_{\\mathrm{L}}$ and others as $0$ for any $i=0,1,\\cdots,Q-1$.  At last, by inputting the detected signal vector into conventional DPIM demodulator, we finally obtain the data bits.  The detailed OSD algorithm for BDPIM is listed in Algorithm 1.\n\\begin{algorithm}[t] \n\\caption{OSD Algorithm for BDPIM}\n\\label{alg:osd}\n\\begin{algorithmic} \n\\STATE  \\textbf{Input:} $\\mathbf{y}$, $L$, $N_s$ and $Q$\n\\STATE Sort $\\mathbf{y}$ and output the order $\\{i(t)\\}$.\n\\STATE Detect $1\\leq i(t)\\leq Q$ as $\\mathrm{A}_{\\mathrm{H}}$ and output the indices $t_1, t_2,\\cdots, t_Q$.\n\\STATE Sort $t_1, t_2,\\cdots, t_Q$ in an ascent order $t_1'< t_2' <\\cdots< t_Q'$.\n\\STATE $t_0=0$.\n\\FOR{$i=0:Q-1$}\n\\STATE Sort $\\mathbf{y}(t_i'+1:t_{i+1}'-1)$ and detect the $(K-1)$ largest as $\\mathrm{A}_{\\mathrm{L}}$ and others as $0$.\n\\ENDFOR\n\\end{algorithmic}\n \\end{algorithm}\n\nAs described, OSD for BDPIM includes two phases. The first involves a sorting algorithm operating on $L$ received signals, whose complexity is $\\mathcal{O}(L\\log L)$. The second involves sorting algorithms operating on $\\mathbf{y}(t_i'+1:t_{i+1}'-1)$ for all $i=0,1,\\cdots,Q-1$. The average length of \n $\\mathbf{y}(t_i'+1:t_{i+1}'-1)$ is around $L\/Q$ and the sorting operations are conducted for $Q$ times. Therefore,  the complexity of the second detection phase is around $\\mathcal{O}\\left(L\\log(L\/Q)\\right)$. In summary, the total complexity of OSD for BDPIM is also around $\\mathcal{O}(L\\log L)$.  \nThe storage delay is in the same order of OSD for DPIM and dependent on the packet size. For large packet detection, the delay may cause unbearable latency. To address this issue, we propose to use a combination of OTD and OSD. OTD is first used to detect $\\mathrm{A}_{\\mathrm{H}}$ according to an optimal threshold $h\\mathrm{A}_{\\mathrm{T}}'$. If a sample is not detected as $\\mathrm{A}_{\\mathrm{H}}$, we store it in a buffer. If it is, we activate OSD to detect the signal sequence in the buffer. In this way, long sequences are split into small pieces. Specifically, the sequence of a packet is divided into the pieces of average length $L\/Q$.\nTherefore, both the storage delay and  processing time can be reduced  by a factor of $Q$. \n\\section{Main Results on BER Performance}\nThis section presents the main results on the approximate upper bounds of the average BER performance of all schemes, including DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD, where the average BER performance is defined by\n\\begin{equation}\nP_e=\\overline{\\mathsf{BER}}P_p,\n\\end{equation}\nwhere $\\overline{\\mathsf{BER}}$ denotes the expected BER in an erroneous packet reflecting the error propagation level and $P_p$ represents the PER. \n\\subsection{Approximate BER Upper Bounds of DPIM-OTD}\n The PER of OTD can be written as\n\\begin{equation}\\label{Pp1}\nP_{p_1}=1-(1-P_c)^L,\n\\end{equation}\nbased on which and the approximate expression of $\\mathrm{A}_{\\mathrm{T}}$ in (\\ref{eqAt}) we have the following theorem:\n\\begin{theorem} \nThe average BER of DPIM-OTD\nis approximately upper bounded by\n\\begin{equation}\\label{Pe1}\n\\begin{split}\n P_{e_1}^\\mathcal{U}\n\\approx\\frac{2-2(1-P_c)^L-LP_c(1-P_c)^{L-1}}{4},\n\\end{split}\n\\end{equation}\nwhere the chip error probability is approximated as\n\\begin{equation}\\label{Pc}\n\\begin{split}\nP_c&\\approx\\frac{L_s-1}{L_s} \\mathcal{Q}\\left(\\frac{h\\sqrt{\\gamma}}{2}+\\frac{1}{h\\sqrt{\\gamma}}\\ln(L_s-1)\\right)\n\\\\&~~~~~+\\frac{1}{L_s} \\mathcal{Q}\\left(\\frac{h\\sqrt{\\gamma}}{2}-\\frac{1}{h\\sqrt{\\gamma}}\\ln(L_s-1)\\right).\n\\end{split}\n\\end{equation}\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix A.\n\\end{IEEEproof}\n\\emph{Remark and Observation:} From (\\ref{Pe1}) and (\\ref{Pc}), it is observed that $P_{e_1}^\\mathcal{U}$ is a function of $\\gamma$, $L$, $h$ and $L_s$.\nBased on the  following inequalities of derivatives\n\\begin{equation}\n\\frac{\\partial P_{e_1}^\\mathcal{U}}{\\partial P_c}>0,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial P_c}{\\partial\\gamma}<0,\n\\end{equation}\nand the chain rule, we have \n\\begin{equation}\n\\frac{\\partial P_{e_1}^\\mathcal{U}}{\\partial \\gamma}=\\frac{\\partial P_{e_1}^\\mathcal{U}}{\\partial P_c}\\cdot\\frac{\\partial P_c}{\\partial\\gamma}<0.\n\\end{equation}\nThis indicates that $P_{e_1}^\\mathcal{U}$ decreases as $\\gamma$ increases. \nTaking the first derivative of $P_{e_1}^\\mathcal{U}$ with respect to $L$, we obtain\n\\begin{equation}\n\\begin{split}\n\\frac{\\partial P_e^\\mathcal{U}}{\\partial L}\\geq0,\n\\end{split}\n\\end{equation}\nwhich shows that $P_{e_1}^\\mathcal{U}$ increases as $L$ increases. \n\n\nWe demonstrate the simulation result of the average BER performance and the derived approximate upper bound of DPIM-OTD in Fig. \\ref{2a1}. In the simulations, $h=1$, $N_s=100$ and $4$-DPIM with $1$ GI are used. \nThe results are averaged over $10^5$ Monte Carlo simulations. Results show that the derived approximate upper bound matches the simulation result well especially in the low and high SNR regime. \n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2a1}\\\\\n \\caption{Approximate BER upper bound of DPIM-OTD where $h=1$, $N_s=100$ and $4$-DPIM with $1$-GI are employed.}\n  \\label{2a1}\n\\end{figure}\n\n\n\n\n\\subsection{Approximate BER Upper Bounds of DPIM-OSD}\nIn use of OSD, chip errors occur in pairs in an erroneous packet. In detail, as long as a \\emph{false alarm error} occurs,  an \\emph{erasure error} occurs, because the number of $\\mathrm{A}$s by OSD is strictly equal to $N_s$. Based on the fact, the PER of DPIM-OSD is indeed the probability of there being at least one pair of chip errors and can be written as\n\\begin{equation}\\label{eqPp2}\nP_{p_2}=\\mathrm{Pr}\\{U_{1:L-N_s}>V_{N_s:N_s}\\},\n\\end{equation}\nwhere $U_{1:L-N_s}$ denotes the $1$st largest of $L-N_s$ received signals when $0$ is transmitted and $V_{N_s:N_s}$ denotes the $N_s$th largest (i.e., the smallest) of $N_s$ received signals when $\\mathrm{A}$ is transmitted.  \nFor any random variables $U$ and $V$, the probability of $U>V$ can be calculated by\n\\begin{equation}\n\\begin{split}\n\\mathrm{Pr}\\{U>V\\}\n&=\\int_{-\\infty}^{+\\infty}[1-F_U(v)]f_V(v) dv,\\\\\n\\end{split}\n\\end{equation}\nwhere $F_U(u)$ is the cumulative distribution function (CDF) of variable $U$ and  $f_V(v)$ is the PDF of variable $V$.\nGiven $U_{k_1:n_1}$ as the $k_1$th largest of $n_1$ real Gaussian variables with mean $\\mu_1$ and variance $\\sigma_n^2\/h^2$,  $V_{k_2:n_2}$ as the $k_2$th largest of $n_2$ real Gaussian variables with mean $\\mu_2$ and variance $\\sigma_n^2\/h^2$, we define an $\\mathcal{OR}$ function to represent $\\mathrm{Pr}\\{U_{k_1:n_1}>V_{k_2:n_2}\\}$ as\n\\setlength{\\arraycolsep}{0.1pt}\n\\begin{equation}\\label{ORF}\n\\begin{split}\n&\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n\\mu_1, & k_1, & n_1 \\\\\n\\mu_2,&k_2, & n_2 \\\\\n\\end{array}\\right.}\\triangleq\\mathrm{Pr}\\{U_{k_1:n_1}>V_{k_2:n_2}\\}\\\\\n& =\\int_{-\\infty}^{+\\infty}[1-F_{U_{k_1:n_1}}(v)]f_{V_{k_2:n_2}}(v) dv.\\\\\n\\end{split}\n\\end{equation}\nAccording to \\cite{Yang2011},  the CDF $F_{U_{k_1:n_1}}(v)$ in (\\ref{ORF}) is given by\n\\begin{equation}\n\\begin{split}\nF_{U_{k_1:n_1}}(v)=\\sum_{k=n_1+1-k_1}^{n_1}\\left(n_1\\atop k\\right)F_U(v)^k(1-F_U(v))^{n_1-k},\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\nF_U(v)=\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{hv-h\\mu_1}{\\sqrt{2}\\sigma_n}\\right)\\right].\n\\end{equation}\nThe PDF $f_{V_{k_2:n_2}}(v)$ is written as \n\\begin{equation}\n\\begin{split}\n&f_{V_{k_2:n_2}}(v)=\\\\\n&\\frac{n_2!}{(n_2-k_2)!(k_2-1)!}F_V(v)^{n_2-k_2}[1-F_V(v)]^{k_2-1}f_{V}(v),\n\\end{split}\n\\end{equation}\nwhere\n\\begin{equation}\nf_V(v)=\\frac{h}{\\sqrt{2\\pi}\\sigma_n}e^{-\\frac{h^2(v-\\mu_2)^2}{2\\sigma_n^2}},\n\\end{equation}\nand \n\\begin{equation}\nF_V(v)=\\frac{1}{2}\\left[1+\\mathrm{erf}\\left(\\frac{hv-h\\mu_2}{\\sqrt{2}\\sigma_n}\\right)\\right].\n\\end{equation}\n\n\n Based on the definition of the $\\mathcal{OR}$ function,\nthe PER of DPIM-OSD can be given by\n\\begin{equation}\\label{Pp2}\nP_{p_2}=\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.},\n\\end{equation}\naccording to which, we derive the following theorem:\n\\begin{theorem}\nThe average BER of DPIM-OSD is approximately upper bounded by\n\\begin{equation}\\label{Pe2a}\n\\begin{split}\n P_{e_2}^\\mathcal{U}\\approx\\frac{1}{6}\\left(\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0,  & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.}+2\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n0, & 2, & L-N_s \\\\\n\\mathrm{A},&N_s-1, & N_s \\\\\n\\end{array}\\right.}\\right),\n\\end{split}\n\\end{equation}\nand the approximation holds when $L$ is large.\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix B.\n\\end{IEEEproof}\n\n\\emph{Remark and Observation:} The approximate upper bound ${P_{e_2}^\\mathcal{U}}$ in (\\ref{Pe2a}) is a function of  $h$, $\\mathrm{A}$, $\\sigma_n$, $L$ and $N_s$. It has more theoretical significance than practical significance since the defined $\\mathcal{OR}$ function involves complicated integrals thus offering no clear insights into the system performance. To simplify the analytical result, we  propose an approximation of the $\\mathcal{OR}$ function as\n\\begin{equation}\n\\begin{split}\n\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n\\mu_1,& k_1, & n_1 \\\\\n\\mu_2,&k_2, & n_2 \\\\\n\\end{array}\\right.}&\\approx\\mathrm{Pr}\\{U_{k_1:n_1}>v_{k_2:n_2}\\}\\\\&=1-F_{U_{k_1:n_1}}(v_{k_2:n_2}),\n\\end{split}\n\\end{equation}\nwhere $v_{k_2:n_2}=\\alpha\\mathbb{E}(V_{k_2:n_2})$ and $\\alpha$ is a constant. We use $\\alpha=0.82$ in this paper and the accuracy of the approximation  is investigated by simulations.  Based on the approximation, we have \n\\begin{equation}\n\\begin{split}\n\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0,  & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.}&\\approx 1-F_U(v_1)^{L-N_s}\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n\\begin{split}\n&\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0,  & 2, & L-N_s \\\\\n\\mathrm{A}, &N_s-1, & N_s \\\\\n\\end{array}\\right.}\n\\approx 1-F_{U_{2:L-N_s}}(v_2)\n\\approx 1-F_{U_{2:L-N_s}}(v_1)\n\\\\&=1-F_U(v_1)^{L-N_s}-\n(L-N_s)F_U(v_1)^{L-N_s-1}[1-F_U(v_1)],\n\\end{split}\n\\end{equation}\nwhere $v_{1}=\\alpha\\mathbb{E}(V_{N_s:N_s})$ and $v_2=\\alpha\\mathbb{E}(V_{N_s-1:N_s})$. Based on these approximations and Theorem 2, we deduce a tractable approximate BER upper bound as\n\\begin{equation}\\label{Pe2b}\n\\begin{split}\n \\overline{P_{e_2}^\\mathcal{U}}\\approx&\\frac{1}{2}\\left[1-F_U(v_1)^{L-N_s}\\right]\\\\&-\\frac{1}{3}(L-N_s)F_U(v_1)^{L-N_s-1}[1-F_U(v_1)].\n\\end{split}\n\\end{equation}\n The new approximate upper bound $\\overline{P_{e_2}^\\mathcal{U}}$ is a function of \n$h$, $\\mathrm{A}$, $v_1$, $\\sigma_n$, $L$ and $N_s$. It is noted that $\\mathbb{E}(V_{N_s:N_s})$ in the expression of $v_1$ is the expectation of the least order statistics of $N_s$ Gaussian samples with mean $\\mathrm{A}$ and covariance $\\sigma_n^2\/h^2$, which can be approximately computed by \\cite{Chen1999}\n\\begin{equation}\n\\mathbb{E}(V_{N_s:N_s})=\\mathrm{A}-\\frac{\\sigma_n\\phi^{-1}\\left(0.5264^{1\/N_s}\\right)}{h},\n\\end{equation}\nwhere $\\phi^{-1}(\\cdot)$ is the inverse of the Gaussian CDF \\cite{Chen1999}. $\\overline{P_{e_2}^\\mathcal{U}}$ is much simpler than (\\ref{Pe2a}) and can be regarded as a closed-form expression because it only involves the $\\mathrm{erf}$ function, which can be further approximated by \\cite{Chiani2003}\n\\begin{equation}\n\\mathrm{erf}(v)\\approx1-\\frac{1}{6}e^{-v^2}-\\frac{1}{2}e^{-\\frac{4}{3}v^2}.\n\\end{equation}\n\n\n\n\nUnder the same simulation setups as that in Section III-A, we simulate the BER performance and compute the derived bounds numerically as illustrated in Fig. \\ref{2a2}.  Results demonstrate that the approximate upper bound in (\\ref{Pe2a}) matches the BER performance well, but is too complicated to analyze.  The tractable approximate upper bound in (\\ref{Pe2b}) has a gap to the simulation result, but it enjoys low complexity.\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2a2}\\\\\n \\caption{Approximate BER upper bounds of DPIM-OSD where $h=1$, $N_s=100$ and $4$-DPIM with $1$-GI are employed.}\n  \\label{2a2}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\setcounter{equation}{42}\n\\begin{equation}\\label{Pe3a}\n\\begin{split}\n P_{e_3}^\\mathcal{U}\n\\approx&\\frac{1}{6}\\left(\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q, & Q \\\\\n\\end{array}\\right.}+2\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  2, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q-1, & Q \\\\\n\\end{array}\\right.}\\right)\\left(1-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}\\right)\\\\\n&+\\frac{1}{6}\\left(\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}+2\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 2, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-2, &K-1 \\\\\n\\end{array}\\right.}\\right)\\left(1-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q, & Q \\\\\n\\end{array}\\right.}\\right)\\\\\n&+\\frac{1}{2}\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q, & Q \\\\\n\\end{array}\\right.}\\times\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}\n\\end{split}\n\\end{equation}\n\n\\begin{equation}\\label{Pe3b}\n\\begin{split}\n\\overline{P_{e_3}^\\mathcal{U}}\n\\approx&\\frac{1}{2}-\\frac{1}{2}F_U(v_3)^{N_s-Q}F_U(v_4)^{\\lfloor KL_s\\rceil-K}-\\frac{1}{3}(N_s-Q)F_U(v_3)^{N_s-Q-1}[1-F_U(v_3)]F_U(v_4)^{\\lfloor KL_s\\rceil-K}\\\\\n&-\\frac{1}{3}(\\lfloor KL_s\\rceil-K)F_U(v_4)^{\\lfloor KL_s\\rceil-K-1}[1-F_U(v_4)]F_U(v_3)^{N_s-Q}\n\\end{split}\n\\end{equation}\n\\hrule\n\\end{figure*}\n\n\n\n\\subsection{Approximate BER Upper Bounds of BDPIM-OSD}\nIn use of BDPIM-OSD, the error probability in the first OSD phase of detecting $\\mathrm{A}_{\\mathrm{H}}$ can be expressed as \n\\setcounter{equation}{40}\n\\begin{equation}\\label{PLH}\nP_{\\mathrm{LH}}=\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, &  1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},& Q, & Q \\\\\n\\end{array}\\right.},\n\\end{equation}\nand that in the second OSD phase of detecting $\\mathrm{A}_{\\mathrm{L}}$ is\n\\begin{equation}\\label{P0L}\nP_{0\\mathrm{L}}=\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}.\n\\end{equation}\nBased on these, we have the following theorem:\n\n\n\n\\begin{theorem}\nThe BER of BDPIM-OSD is \napproximately upper bounded by (\\ref{Pe3a}) when $L$ is large.\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix C.\n\\end{IEEEproof}\n\n\\emph{Remark and Observation:} It is observed from (\\ref{Pe3a}) that $ P_{e_3}^\\mathcal{U}$ is affected by various system parameters such as $\\mathrm{A}_\\mathrm{L}$, $\\mathrm{A}_\\mathrm{H}$, $K$ as well as the channel condition $h$ and $\\sigma_n$. To simplify its expression, based on the approximation of the $\\mathcal{OR}$ function used in Section III-B, we obtain a tractable approximate BER upper bound as (\\ref{Pe3b}), where $v_3=\\alpha\\mathbb{E}(V_{Q:Q})$ and $v_4=\\alpha\\mathbb{E}(V_{K-1:K-1})$.\n\n\n\nTo corroborate our studies, we simulate a system using BDPIM with 1 GI and OSD. The parameters are set as: $h=1$, $N_s=100$, $K=10$, $\\mathrm{A}_{\\mathrm{H}}=2.3$ and $\\mathrm{A}_{\\mathrm{L}}=0.86$. The simulation result and the bounds are given in  Fig. \\ref{2a3}. It is observed that the approximate upper bound in (\\ref{Pe3a}) is tight over all SNR regimes and the tractable  approximate upper bound in (\\ref{Pe3b}) is tight only in the low and high SNR regime. From these results, the accuracy in approximating the $\\mathcal{OR}$ function can also be validated. \n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2a3}\\\\\n \\caption{Approximate BER upper bounds of BDPIM-OSD where $h=1$, $N_s=100$, $K=10$, $\\mathrm{A}_{\\mathrm{H}}=2.3$, $\\mathrm{A}_{\\mathrm{L}}=0.86$ and $4$-BDPIM with $1$GI are employed.}\n  \\label{2a3}\n\\end{figure}\n\n\n\\subsection{Approximate BER Upper Bounds of BDPIM-OTD-OSD}\n\n\nFor BDPIM-OTD-OSD, the first phase is detecting the barrier chip of amplitude $\\mathrm{A}_{\\mathrm{H}}$ by using a  threshold $h\\mathrm{A}_\\mathrm{T}'$. Based on the similar analysis in Section III-A, $\\mathrm{A}_\\mathrm{T}'$ can be computed by\n\\setcounter{equation}{44}\n\\begin{equation}\\label{eqAt1}\n\\mathrm{A}_{\\mathrm{T}}'=\\frac{\\mathrm{A}_\\mathrm{H}+\\mathrm{A}_\\mathrm{L}}{2}+\\frac{\\sigma_n^2}{h^2(\\mathrm{A}_\\mathrm{H}-\\mathrm{A}_\\mathrm{L})}\\ln(K-1).\n\\end{equation}\nUsing $h\\mathrm{A}_{\\mathrm{T}}'$ for detection, the PEP in the first OTD phase  can be calculated by \n\\begin{equation}\\label{PLHN}\nP_{\\mathrm{LH}}'=1- (1-P_{c}')^{N_s},\n\\end{equation}\nwhere $P_c'$ writes \n\\begin{equation}\\label{Ps}\n\\begin{split}\nP_{c}'=&\\frac{1}{K} \\mathcal{Q}\\left(\\frac{h(\\mathrm{A}_{\\mathrm{H}}-\\mathrm{A}_{\\mathrm{L}})}{2\\sigma_n}+\\frac{\\sigma_n}{h(\\mathrm{A}_{\\mathrm{H}}-\\mathrm{A}_{\\mathrm{L}})}\\ln(K-1)\\right)\n\\\\&+\\frac{K-1}{K} \\mathcal{Q}\\left(\\frac{h(\\mathrm{A}_{\\mathrm{H}}-\\mathrm{A}_{\\mathrm{L}})}{2\\sigma_n}-\\frac{\\sigma_n}{h(\\mathrm{A}_{\\mathrm{H}}-\\mathrm{A}_{\\mathrm{L}})}\\ln(K-1)\\right).\n\\end{split}\n\\end{equation}\nThe error detection probability in the second OSD phase is the same as that in (\\ref{P0L}). Based on these, we have the following theorem:\n\\begin{theorem}\nThe BER of BDPIM-OTD-OSD is approximately upper bounded by (\\ref{Pe4a}) when $L$ is large.\n\\end{theorem}\n\\begin{IEEEproof}\nSee Appendix D.\n\\end{IEEEproof}\n\\begin{figure*}\n\\setcounter{equation}{47}\n\\begin{equation}\\label{Pe4a}\n\\begin{split}\n P_{e_4}^\\mathcal{U}\n\\approx&\\frac{2-2(1-P_c')^{N_s}-LP_c'(1-P_c')^{N_s-1}}{4}\\left(1-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}\\right)\\\\\n&+\\frac{1}{6}\\left(\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}+2\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 2, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-2, &K-1 \\\\\n\\end{array}\\right.}\\right)(1-P_c')^{N_s}\\\\\n&+\\frac{1}{2}\\left[1-(1-P_c')^{N_s}\\right]\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}\n\\end{split}\n\\end{equation}\n\n\\begin{equation}\\label{Pe4b}\n\\begin{split}\n\\overline{P_{e_4}^\\mathcal{U}}\n\\approx&\\frac{1}{2}-\\frac{1}{2}(1-P_c')^{N_s}F_U(v_4)-\\frac{1}{4}LP_c'(1-P_c')^{N_s-1}F_U(v_4)^{\\lfloor KL_s\\rceil-K}\\\\\n&-\\frac{1}{3}(\\lfloor KL_s\\rceil-K)F_U(v_4)^{\\lfloor KL_s\\rceil-K-1}[1-F_U(v_4)](1-P_c')^{N_s}\n\\end{split}\n\\end{equation}\n\\hrule\n\\end{figure*}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2a4}\\\\\n \\caption{Approximate BER upper bounds of BDPIM-OTD-OSD where $h=1$, $N_s=100$, $K=10$, $\\mathrm{A}_{\\mathrm{H}}=2.3$, $\\mathrm{A}_{\\mathrm{L}}=0.86$ and $4$-BDPIM with $1$GI are employed.}\n  \\label{2a4}\n\\end{figure}\n\n\\emph{Remark and Observation:}\nSimilarly, using the approximation of the $\\mathcal{OR}$ function used in Section III-B, we obtain a tractable approximate BER upper bound as (\\ref{Pe4b}).\n Under the same simulation setups as that in Section III-C, we simulate BDPIM-OTD-OSD. The simulation result and the bounds are given in  Fig. \\ref{2a4}. Similarly, it is also observed that the upper bound in (\\ref{Pe4a}) are quite tight over all SNR regimes and that in (\\ref{Pe4a}) is close to the BER performance in the low and high SNR regime.\n\n\n\n\n\\section{Simulation and Discussion}\nThis section presents the simulation results and has four subsections. In the first subsection, we show the superiority of the proposed OSD over OTD and MLSD.  In the second subsection,  we show the superiority of BDPIM in uncoded and coded systems. In the third subsection, we study the impact of the parameters on the performance of the proposed BDPIM. In the last subsection, we investigate the extension to Gamma-Gamma turbulence channels.\n\n\n\\subsection{Superiority of OSD}\nFirstly, we compare the proposed OSD with OTD and MLSD using exhaustive search in terms of BER and time complexity. Because MLSD using exhaustive search has prohibitive high complexity when the sequence is long, we compare these schemes in a system transmitting small packets containing $4$ symbols per packet. In the comparison, $h=1$ and $4$-DPIM with $1$ GI are employed. We run $10^5$ Monte Carlo simulations and evaluate the BER performance as well as the average time complexity in UTs. The BER performance comparison is shown in Fig. \\ref{MLSDa}, from which we observe that OSD exhibits the same performance as MLSD using exhaustive search and considerably outperforms OTD over a wide SNR regime. The average time complexity  is included in Fig. \\ref{MLSDb}, which shows that OSD is much more computationally efficient than MLSD using exhaustive search.\n\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig0_a}\\\\\n \\caption{Comparisons among OTD, OSD and MLSD with exhaustive search in terms of BER  where $N_s=4$ and $4$-DPIM with $1$ GI are employed.}\n  \\label{MLSDa}\n\\end{figure}\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig0_b}\\\\\n \\caption{Comparisons among OTD, OSD and MLSD in terms of time complexity, where a UT is the amount of time it takes for a multiplication.}\n  \\label{MLSDb}\n\\end{figure}\n\n\n\\subsection{Superiority of BDPIM}\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2b}\\\\\n \\caption{Comparisons among MDPIM-OTD \\cite{Ghassemlooy2006}, DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD in terms of uncoded BER performance.}\n  \\label{BDPIM}\n\\end{figure}\n\nSecondly, we compare BDPIM with DPIM and MDPIM \\cite{Ghassemlooy2006} using various detectors in uncoded and coded systems. The parameters $h=1$, $N_s=100$, $K=10$, and $4$-ary BDPIM with $1$ GI are employed. In coded systems, an interleaver of length $K\\log M=20$ is used. To show the proposed BDPIM in association with high-rate FEC codes, a rate-$1\/2$ convolution code is adopted and its code generator can be expressed as $[g_0,g_1]^T=[1+d+d^2,1+d^2]$. Additionally, the Viterbi hard-decision decoder is used for decoding. For $\\mathrm{A}_{\\mathrm{H}}$ and $\\mathrm{A}_{\\mathrm{L}}$, we adopt the optimal solution obtained in Section IV-C. The uncoded simulation results are included Fig. \\ref{BDPIM}. It demonstrates that BDPIM-OSD exhibits the best BER performance in all depicted SNR regimes. It slightly outperforms BDPIM-OTD-OSD, which is applicable in large packet cases. Compared with DPIM-OSD, BDPIM-OSD offers a significant gain at the medium SNR regime. The gain shrinks as SNR goes larger. The reason is as follows. The BER performance is affected by two factors, the PER and the expected BER (affected by error propagation) in an erroneous packet. BDPIM uses some power to offer built-in barriers, which slightly increases PER and greatly reduces the expected BER in an erroneous packet. At the medium SNR regime, the second factor affects the system performance more. As SNR increases, the impacts of two factors on BER performance become comparable. Therefore, the gain shrinks as SNR increases. From Fig. \\ref{BDPIM}, we also observe that BDPIM-OSD greatly outperforms DPIM-OTD by $2$ dB at a BER of $10^{-2}$. Additionally, we compare BDPIM with the MDPIM in \\cite{Ghassemlooy2006}, which uses multiple amplitudes to additionally transmit information. Note that as the average symbol duration of MDPIM is shorter than BDPIM (shown in Table II), we simulate MDPIM under the same average power per chip for fairness. It is observed that BDPIM-OSD considerably outperforms MDPIM by more than $4$ dB. This is reasonable because MDPIM targets to improve the transmission rate while BDPIM aims to enhance the system reliability.\n\n\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig2c}\\\\\n \\caption{Comparisons among MDPIM-OTD \\cite{Ghassemlooy2006}, DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD in terms of coded BER performance.}\n  \\label{codedBDPIM}\n\\end{figure}\n\nThe comparison made in coded systems is demonstrated in Fig. \\ref{codedBDPIM}.  It shows that BDPIM-OSD still achieves the best performance and is slightly better than BDPIM-OTD-OSD. Both of them greatly outperform the others. Note that the gain brought by BDPIM-OSD does not shrink as SNR increases in coded systems compared with DPIM-OSD. On the contrary, the gain goes larger as SNR increases. This is because the length of block error is short in BDPIM by limiting the error propagation which enables the functionality of the rate-1\/2 FEC code in improving BER performance. For DPIM, even though its PER is low in the high SNR regime (shown in uncoded systems), the error bits in an erroneous packet are too many to be corrected. As a result, the gain becomes more significant. Moreover, Fig. \\ref{codedBDPIM} also demonstrates that the gains brought by BDPIM-OSD  are also more considerable than that in Fig. \\ref{BDPIM} compared with DPIM-OTD and MDPIM-OTD.\nTherefore, we conclude that the gain brought by BDPIM is more considerable in coded systems. \n\n\\subsection{Parameter Impact on the Performance of BDPIM}\nGiven the number of symbols contained in a packet $N_s$, the parameters of BDPIM include $K$, $\\mathrm{A}_{\\mathrm{L}}$ and $\\mathrm{A}_{\\mathrm{H}}$.  The parameter optimization problems of BDPIM with different detectors can be expressed as\n\\begin{equation}\\label{Problem}\n\\begin{split}\nK^*,\\mathrm{A}_{\\mathrm{L}}^*, \\mathrm{A}_{\\mathrm{H}}^*=&\\arg\\min_{K,\\mathrm{A}_{\\mathrm{L}},\\mathrm{A}_{\\mathrm{H}}} P_{e_i}^{\\mathcal{U}}(K,\\mathrm{A}_{\\mathrm{L}},\\mathrm{A}_{\\mathrm{H}}), ~i=3,4\\\\\n\\mathrm{subject~to:~}& 0<\\mathrm{A}_{\\mathrm{L}}<\\mathrm{A}_{\\mathrm{H}}\\textrm{~and~} (\\ref{Con1}).\\\\\n\\end{split}\n\\end{equation}\nGiven $K$ and $\\mathrm{A}$, the constraint in (\\ref{Con1}) can be expressed as \n\\begin{equation}\n\\mathrm{A}_{\\mathrm{H}}=K\\mathrm{A}-(K-1)\\mathrm{A}_{\\mathrm{L}}.\n\\end{equation}\nThus, the problem in (\\ref{Problem}) becomes a one-dimensional search problem of finding the optimal $\\mathrm{A}_{\\mathrm{L}}$ from $(0,\\mathrm{A})$. As the first derivative of $P_{e_i}^{\\mathcal{U}}(K,\\mathrm{A}_{\\mathrm{L}},\\mathrm{A}_{\\mathrm{H}}), ~i=3,4$ with respect to $\\mathrm{A}_{\\mathrm{L}}$ has no simple expression, it is difficult to derive the closed-form solution. Therefore, we resort to using numerical search methods such as the bisection search algorithm \\cite{Boyd2004} in this paper. \n\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig3a_new}\\\\\n \\caption{SNR thresholds for a BER of $10^{-3}$ versus different $\\mathrm{A}_{\\mathrm{L}}\/\\mathrm{A}$.}\n  \\label{3a_new}\n\\end{figure}\n\n\n\n\nTo show how these parameters affect the performance of BDPIM, we simulate the system with a varying parameter. By setting $\\mathrm{A}_{\\mathrm{L}}\/\\mathrm{A}$ to vary from $0$ to $1$ by a step of $0.01$, $h=1$, $N_s=100$ and $K=10$, we obtain the SNR thresholds for a BER of $10^{-3}$ in coded and uncoded BDPIM systems as shown in Fig. \\ref{3a_new}.   It shows that the optimal value of $\\mathrm{A}_{\\mathrm{L}}$ in coded systems is smaller than that in uncoded systems. That is, the optimal value of $\\mathrm{A}_{\\mathrm{H}}$ in coded systems is larger than that in uncoded systems. It can be explained as follows. The FEC code can correct the errors of the second phase conditioned the first phase is correctly detected. Therefore, the larger $\\mathrm{A}_{\\mathrm{H}}$ should be adopted in coded systems to ensure the correctness of the first detection phase.\nAdditionally, by setting $K$ to vary in $\\{5,10,20,25,50\\}$, $h=1$, $N_s=100$ and $\\mathrm{A}_{\\mathrm{L}}$, $\\mathrm{A}_{\\mathrm{H}}$ as the searched optimal values, we obtain the SNR thresholds for a BER of $10^{-3}$ in coded and uncoded BDPIM systems as shown in Fig. \\ref{3b_new}. It shows that the values of $K$ affect coded systems more than uncoded systems and the optimal ones for all systems are $10$. Specifically, the SNR thresholds vary from $16.5$ dB to $17.3$ dB in uncoded systems as $K$ varies, while the SNR thresholds vary in a much larger range from $14.8$ dB to $16.8$ dB in coded systems as $K$ varies. This is because $K$ affects the length of block error and determines whether the block error can be corrected or not.\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig3b_new}\\\\\n \\caption{SNR thresholds for a BER of $10^{-3}$ versus different $K$.}\n  \\label{3b_new}\n\\end{figure}\n\\subsection{Extension to Gamma-Gamma Turbulence Channels}\nIn this part, we consider the extension of the proposed designs to other channels. The Gamma-Gamma turbulence channels are taken as an example. The ergodic  approximate  upper bound for any given  channel coefficient distribution can be expressed as\n\\begin{equation}\\label{EPe}\n\\widetilde{P_{e_{i}}^{\\mathcal{U}}}=\\int_{0}^{+\\infty}P_{e_{i}}^{\\mathcal{U}}(h)f_{h}{(h)} dh, ~i=1,2,3,4.\n\\end{equation}\nFor Gamma-Gamma turbulence channels, the channel coefficient distribution can be expressed as\\cite{Jaiswal2017}\n\\begin{equation}\\label{pdfH}\nf_{h}{(h)}=\\frac{2(\\lambda\\mu)^{\\frac{\\lambda+\\mu}{2}}}{\\Gamma(\\lambda)\\Gamma(\\mu)}h^{\\frac{\\lambda+\\mu}{2}}K_{\\lambda-\\mu}\\left(\\sqrt{\\lambda\\mu h}\\right)\n\\end{equation}\nwhere $\\Gamma(\\cdot)$ and $K_{\\lambda-\\mu}(\\cdot)$ are the Gamma function and the modified Bessel function, respectively. In (\\ref{pdfH}),  $\\lambda$ and $\\mu$ are the turbulence parameters that characterize the irradiance fluctuations.\n By inserting (\\ref{pdfH}) into (\\ref{EPe}), one can readily extend to the analytical results given in Theorems 1-4 to Gamma-Gamma turbulence channels. \n\nWe simulate the system under weak turbulence condition, where $\\lambda$ and $\\mu$ are set as $11.6$ and $10.1$ \\cite{Jaiswal2017}, respectively. The  results are included in Fig. \\ref{GGBDPIM}. Results show that the proposed BDPIM and OSD can bring about $4$-dB gain compared with the widely adopted DPIM-OTD over a wide SNR regime.\n\n \\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig4a}\\\\\n \\caption{Comparisons among MDPIM-OTD \\cite{Ghassemlooy2006}, DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD in terms of BER performance over Gamma-Gamma weak turbulence channel.}\n  \\label{GGBDPIM}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusions}\nThis paper proposed an OSD and a BDPIM to improve the system reliability of DPIM. With  low complexity, OSD achieves the same system performance as the optimal MLSD in terms of BER. Through specific power allocation, BDPIM makes the error propagation bounded and enables the association with high-rate FEC codes, which greatly improves the system reliability. The  approximate BER upper bounds of four schemes including DPIM-OTD, DPIM-OSD, BDPIM-OSD and BDPIM-OTD-OSD were derived. The approximate upper bounds were shown tight over all SNR regimes, but they are expressed with complicated integrals. To fix this, this paper also provided tractable approximate upper bounds, which are expressed in closed-form expressions. We studied how parameter settings affect the performance by simulations.  We compared all schemes over AWGN channel and over Gamma-Gamma turbulence channels. Result showed that the proposed BDPIM and OSD can bring considerable performance gain and enhance the system reliability significantly.\n\n\n\n\n\\appendices\n\n\n\n\\section{Proof of Theorem 1}\nFor DPIM-OTD, the expected BER in an erroneous packet  can be expressed as\n\\begin{equation}\n\\overline{\\mathsf{BER}}_1=\\beta_{1}\\overline{\\mathsf{BER}}_{1,1}+(1-\\beta_{1})\\overline{\\mathsf{BER}}_{1,o}.\n\\end{equation}\nwhere $\\beta_1$ is the probability of there being a chip error conditioned a packet is wrongly detected, $\\overline{\\mathsf{BER}}_{1,1}$ represents the expected BER in an erroneous packet with a single-chip error and $\\overline{\\mathsf{BER}}_{1,o}$ stands for the expected BER in an erroneous packet with more than one chip error.\nFor either a \\emph{false alarm error} or an \\emph{erasure error} in OTD, the error not only corrupts the bits directly associated with that chip, but also shifts the bits that follow those bits, such that\n\\begin{equation}\\label{BER1}\n\\overline{\\mathsf{BER}}_{1,1}=\\sum_{t=1}^{L}\\frac{N_s-\\mathcal{N}(t)}{2LN_s},\n\\end{equation}\nwhere $t$ is the error position and $\\mathcal{N}(t)$ stands for the number of symbols (i.e., the number of $\\mathrm{A}$s)  between the $1$st and the $t$th chip. Since the average symbol duration is $L_s$, $\\mathcal{N}(t)\\approx t\/L_s$ and $L\\approx N_sL_s$. Substituting these into (\\ref{BER1}), we can get $\\overline{\\mathsf{BER}}_{1,1}\\approx (L-1)\/4L\\approx1\/4$. \nFor the other cases,  $\\overline{\\mathsf{BER}}_{1,o}$ is less than $1\/2$ (the BER of random guess).\nAs to the probability $\\beta_1$, it can be calculated by\n\\begin{equation}\\label{beta1}\n\\begin{split}\n\\beta_1=\\frac{LP_c(1-P_c)^{L-1}}{1-(1-P_c)^L}.\n\\end{split}\n\\end{equation}\nSummarizing above, $\\overline{\\mathsf{BER}}_1\\leq \\frac{2-\\beta_1}{4}$ and $P_{e_1}\\leq \\frac{2-\\beta_1}{4}P_{p_1}$. By substituting $\\beta_1$ in (\\ref{beta1}) and $P_{p_1}$ in (\\ref{Pp1}), Theorem 1 is proved.\n\n\n\n\n\n\n\\section{Proof of Theorem 2}\nAs stated in Section III-B, the chip errors of OSD occur in pairs.\nFor DPIM-OSD, the expected BER in an erroneous packet  can be expressed as\n\\begin{equation}\n\\overline{\\mathsf{BER}}_2=\\beta_{2}\\overline{\\mathsf{BER}}_{2,1}+(1-\\beta_{2})\\overline{\\mathsf{BER}}_{2,o}.\n\\end{equation}\nwhere $\\beta_2$ represents the probability of there being a pair of chip errors conditioned a packet is wrongly detected, $\\overline{\\mathsf{BER}}_{2,1}$ denotes the expected BER in an erroneous packet with a pair of chip errors and $\\overline{\\mathsf{BER}}_{2,o}$ is the expected BER in an erroneous packet with more than one pair of chip errors.\nWhen a single pair of chip errors occur in OSD, the errors will corrupt the bits between these two chips and $\\overline{\\mathsf{BER}}_{2,1}$ can be expressed as\n\\begin{equation}\n\\overline{\\mathsf{BER}}_{2,1}=\\sum_{t_1=1}^{L}\\sum_{t_2=t_1+1}^{L}\\frac{\\mathcal{N}(t_2)-\\mathcal{N}(t_1)}{L(L-1)N_s},\n\\end{equation}\nwhere $t_1$ and $t_2$ are the error positions. Based on the approximations $\\mathcal{N}(t_2)\\approx t_2\/L_s$, $\\mathcal{N}(t_1)\\approx t_1\/L_s$ and $L\\approx N_sL_s$, the BER of DPIM-OSD can be obtained as\n\\begin{equation}\n\\overline{\\mathsf{BER}}_{2,1}\\approx \\frac{2L^2-7L+5}{12L^2}\\approx \\frac{1}{6}.\n\\end{equation}\nThis indicates that averagely, one sixth of bits are in error in an erroneous packet for this case. \nFor the other cases, $\\overline{\\mathsf{BER}}_{2,o}$ is less than $1\/2$ (the BER of random guess), i.e., $\\overline{\\mathsf{BER}}_{2,o}\\leq \\frac{1}{2}$.\nAs to the probability $\\beta_2$, it can be expressed as\n\\begin{equation}\\label{beta2}\n\\begin{split}\n\\beta_2&=\\frac{P_{p_2}-\\mathrm{Pr}\\{U_{2:L-N_s}>V_{N_s-1:N_s}\\}}{P_{p_2}}\\\\\n&\\approx\\frac{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n0, & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.}-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n0, & 2, & L-N_s \\\\\n\\mathrm{A},&N_s-1, & N_s \\\\\n\\end{array}\\right.}}{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n0, & 1, & L-N_s \\\\\n\\mathrm{A},&N_s, & N_s \\\\\n\\end{array}\\right.}},\n\\end{split}\n\\end{equation}\nbecause $P_{p_2}$ in (\\ref{eqPp2}) is the probability that there are at least one pair of chip errors in an erroneous packet, $\\mathrm{Pr}\\{U_{2:L-N_s}>V_{N_s-1:N_s}\\}$ represents the probability that there are at least two pairs of chip errors in an erroneous packet and the difference between the two terms is the probability of a single pair of chip errors.\nSummarizing above, $\\overline{\\mathsf{BER}}_2\\leq \\frac{(3-2\\beta_2)}{6}$ and $P_{e_2}\\leq \\frac{(3-2\\beta_2)}{6}P_{p_2}$. By substituting $\\beta_2$ in (\\ref{beta2}) and $P_{p_2}$ in (\\ref{Pp2}), Theorem 2 is proved.\n\n\n\n\n\n\n\\section{Proof of Theorem 3}\nThere are 5 events for all error cases of BDPIM-OSD.\n\nEvent 1: The first OSD phase is wrongly detected, the second OSD phase is correctly detected and there are only a single pair of chip errors in the first OSD phase. The probability  of this event is $\\beta_{3,a}P_{\\mathrm{LH}}(1-P_{0\\mathrm{L}})$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,1}$. According to the similar analysis in Appendix B, $\\overline{\\mathsf{BER}}_{3,1}\\approx1\/6$ and $\\beta_{3,a}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n&\\beta_{3,a}=\\frac{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, & 1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},&Q, & Q \\\\\n\\end{array}\\right.}-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, & 2, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},&Q-1, & Q \\\\\n\\end{array}\\right.}}{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{cccc}\n\\mathrm{A}_{\\mathrm{L}}, & 1, & N_s-Q \\\\\n\\mathrm{A}_{\\mathrm{H}},&Q, & Q \\\\\n\\end{array}\\right.}}.\\\\\n\\end{split}\n\\end{equation}\n\n\nEvent 2: The first OSD phase is correctly detected, the second OSD phase is wrongly detected and there are only a single pair of of chip errors in the second OSD phase. The probability of this event is $\\beta_{3,b}(1-P_{\\mathrm{LH}})P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,2}$. According to the similar analysis in Appendix B, $\\overline{\\mathsf{BER}}_{3,2}\\approx1\/6$ and $\\beta_{3,b}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n&\\beta_{3,b}=\\frac{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 2, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-2, &K-1 \\\\\n\\end{array}\\right.}}{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}},\\\\\n\\end{split}\n\\end{equation}\n\nEvent 3: The first OSD phase is wrongly detected, the second OSD phase is correctly detected and there are more than one pair of chip errors in the first OSD phase. The probability  of this event is $(1-\\beta_{3,a})P_{\\mathrm{LH}}(1-P_{0\\mathrm{L}})$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,3}$, which is less than $1\/2$ (the BER of random guess). \n\nEvent 4: The first OSD phase is correctly detected, the second OSD phase is wrongly detected and there are more than one pair of chip errors in the second OSD phase. The probability  of this event is $(1-\\beta_{3,b})(1-P_{\\mathrm{LH}})P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,4}$, which is less than $1\/2$ (the BER of random guess). \n\nEvent 5: Both OSD phases are wrongly detected. The probability  of this event is $P_{\\mathrm{LH}}P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{3,5}$, which is less than $1\/2$ (the BER of random guess). \n\nSummarizing all above and substituting $P_{\\mathrm{LH}}$ in (\\ref{PLH}) and, $P_{0\\mathrm{L}}$ in (\\ref{P0L}), Theorem 3 can be derived.\n\n\n\\section{Proof of Theorem 4}\nThere are 5 events for all error cases of BDPIM-OTD-OSD.\n\nEvent 1: The first OTD phase is wrongly detected, the second OSD phase is correctly detected and there is only a single chip error in the first OTD phase. The probability  of this event is $\\beta_{4,a}P_{\\mathrm{LH}}'(1-P_{0\\mathrm{L}})$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,1}$. According to the similar analysis in Appendix A, $\\overline{\\mathsf{BER}}_{4,1}\\approx1\/4$ and $\\beta_{4,a}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n&\\beta_{4,a}=\\frac{N_sP_c'(1-P_c')^{N_s-1}}{1-(1-P_c')^N_s}.\n\\end{split}\n\\end{equation}\n\n\nEvent 2: The first OTD phase is correctly detected, the second OSD phase is wrongly detected and there are only a single pair of of chip errors in the second OSD phase. The probability of this event is $\\beta_{4,b}(1-P_{\\mathrm{LH}}')P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,2}$. According to the similar analysis in Appendix B, $\\overline{\\mathsf{BER}}_{4,2}\\approx1\/6$ and $\\beta_{4,b}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n&\\beta_{4,b}=\\frac{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}-\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 2, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-2, &K-1 \\\\\n\\end{array}\\right.}}{\\mathcal{OR}{\\scriptsize\\left|\\begin{array}{ccc}\n0, & 1, & \\lfloor KL_s\\rceil-K\\\\\n\\mathrm{A}_{\\mathrm{L}},&K-1, &K-1 \\\\\n\\end{array}\\right.}},\\\\\n\\end{split}\n\\end{equation}\n\nEvent 3: The first OTD phase is wrongly detected, the second OSD phase is correctly detected and there are more than one chip error in the first OTD phase. The probability  of this event is $(1-\\beta_{4,a})P_{\\mathrm{LH}}'(1-P_{0\\mathrm{L}})$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,3}$, which is less than $1\/2$ (the BER of random guess). \n\nEvent 4: The first OTD phase is correctly detected, the second OSD phase is wrongly detected and there are more than one pair of chip errors in the second OSD phase. The probability  of this event is $(1-\\beta_{4,b})(1-P_{\\mathrm{LH}}')P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,4}$, which is less than $1\/2$ (the BER of random guess). \n\nEvent 5: Both OTD and OSD phases are wrongly detected. The probability  of this event is $P_{\\mathrm{LH}}'P_{0\\mathrm{L}}$ and the expected BER in a packet of this event is denoted by  $\\overline{\\mathsf{BER}}_{4,5}$, which is less than $1\/2$ (the BER of random guess). \n\nSummarizing all above and substituting $P_{\\mathrm{LH}}'$ in (\\ref{PLHN}) and, $P_{0\\mathrm{L}}$ in (\\ref{P0L}), Theorem 4 can be derived.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nProduction of heavy quarks is of substantial and ongoing interest in high energy hadronic collisions. This statement had not changed in nearly 40 years, when the charm and bottom flavoured particles were discovered. The energy scale for the production of charm and bottom quarks is significantly higher than the typical Quantum Chromodynamics (QCD) scale, $\\Lambda_{QCD}\\sim 0.2$ GeV. This gives the value of strong coupling of the order of $\\alpha_S \\sim 0.2 - 0.3$, which is small enough to apply perturbative QCD techniques. At the present level of knowledge and current experimental abilities, heavy flavours are known as one of the best testing grounds of the theory of hard QCD interactions. Theoretical analyses of charm and bottom cross sections in proton-proton interactions provide\nan unique precision tool in this context. Due to their mass, heavy quarks are also belived to be a special probe of the medium created in heavy ion collisions.\nSince heavy quarks are only produced in the initial stage of the heavy ion collisions, heavy quark distributions from proton-proton interactions\nsupply a well defined initial state. Their further propagation through the hot and dense medium probes its interesting characteristics.\n \nMeasurements of charm and bottom cross sections at hadron colliders can be performed in the so-called indirect way.\nThis method is based on measurement of leptons from semileptonic decays of open charm and bottom mesons, which are often called non-photonic. The semileptonic decay modes allow for an indirect measurement of the D and B meson cross sections and by further extrapolation the charm and bottom quark cross sections. The indirect methodologies to measure non-photonic electrons\/muons have been used since the early 1970's \\cite{Appel:1974yu}.\n\nThe decay of hadrons by the weak interaction can be viewed as a process of decay of their constituent quarks. \nThe charm and bottom flavours are not preserved in weak interactions, so their weak decays are possible. Within the semileptonic decays, the allowed (according to electric charge conservation) transitions $b \\to c,u$ and $c \\to s,d$ are involved by the emission of charged $W$ boson, which further creates a lepton and the corresponding antineutrino $W \\to l \\bar{\\nu}_l$. The semileptonic decay widths are proportional to the square of the appropriate elements of the Cabbibo-Kobayashi-Maskawa (CKM) quark-mixing matrix \\cite{Kobayashi:1973fv}, which contains information on the strength of flavour-changing\nweak decays. The decays within the same quark generation are strongly favoured over decays between generations.\n\nMain virtues of the semileptonic modes come from the fact that they have bigger branching fractions than the hadronic $D$ meson decay channels. Moreover, in the former the effects of strong interactions can be isolated and thus they are better accessible\nexperimentally. In addition, direct lepton production through weak and electromagnetic interactions in hadronic collisions is suppressed relative to the strong interaction by twelve and four orders of magnitude, respectively \\cite{Halzen:1984mc}.\n\nHowever, electrons and muons are certainly not rare paticles because they are abundantly produced in light hadron decays.\nThese rather problematic background must be accounted for and eliminated through experimental techniques. If the primary sources of background are well understood and substracted, the remaining events can then be attributed to the heavy flavour signal.\n\nAnother method for experimental investigation of charm and bottom quark production at hadron colliders is the direct procedure\nbased on full reconstruction of all decay products of open charm and bottom mesons, for instance in the \n$D^0 \\to K^- \\pi^+ $, $D^+ \\to K^- \\pi^+ \\pi^+$ or $B^+ \\to J\/\\psi K^+ \\to K^+ \\mu^+ \\mu^-$ channels. The hadronic decay products can be used to built\ninvariant mass distributions, permitting direct observation of $D$ or $B$ meson as a peak in the experimental invariant mass spectrum. Open charm $D$ and $B$ mesons are characterized by rather long lifetimes, of the order of $\\sim 10^{-13}$ and $\\sim 10^{-12}$ seconds, respecitvely. Charm and bottom quarks decay essentially at the collision vertex, while heavy flavour mesons decay from a secondary vertex offset by the boosted lifetime of the paticle. In the direct approach the charm and bottom contributions can be well separated, which is not the case in the indirect method. In the latter case, it can be achieved only within the analysis of lepton-meson (e.g. $e$-$D$) correlations \\cite{Mischke:2008af}, which are easily available e.g. in the $k_t$-factorization approach.\n \nThe STAR and PHENIX collaborations have measured transverse momentum distributions of electrons coming from the semileptonic decays of\nheavy flavoured hadrons in proton-proton scattering at the RHIC energy $\\sqrt{s} = 200$ GeV with lepton transverse momenta up to $10$ GeV in the midrapidity region \\cite{Adare:2006hc,Abelev:2006db}. In addition, the STAR collaboration was able to separate the charm and bottom contributions to the spectra of heavy flavour electrons \\cite{Agakishiev:2011mr}.\nThe PHENIX collaboration has also measured non-photonic dilepton invariant mass spectrum from $0$ to $8$ GeV in proton-proton collisions at $\\sqrt{s}=200$ GeV \\cite{Adare:2008ac}. On the theoretical side, the cross sections for inclusive production of the non-photonic electrons at RHIC have been studied theoretically up to the next-to-leading order pQCD collinear approximation within the FONLL approach in Ref.~\\cite{Cacciari:2005rk}. The first theoretical investigation within the competitive QCD $k_t$-factorization framework was done in Refs.~\\cite{Luszczak:2008je,Maciula:2010yw}, including more exclusive studies of kinematical correlations.\n\nVery recently, the STAR collaboration measured for the first time transverse momentum spectra of $D^*$ and $D^0$ mesons up to $6$ GeV at $\\sqrt{s} = 200$ \\cite{Adamczyk:2012af} and $500$ GeV \\cite{STAR-Dmeson500} in proton-proton collisions. Before, studies of charm production at RHIC through hadronic decay channels were performed only in Cu-Cu collisions \\cite{Baumgart:2008zz} and in proton-proton scattering but for $D^*$ mesons produced in jets \\cite{Abelev:2009aj}.\nUp to now, the new STAR proton-proton data on open charm production were studied only in the context of high energy pA collisions in the Color Glass Condensate framework with the unintegrated gluon densities from the solution of rcBK equation \\cite{Fujii:2013yja}.   \n\nOur aim here is to make first theoretical analysis of the measured hadron-level charm differential cross section within the $k_t$-factorization approach. The open charm meson data allow us to make more direct comparison of the theoretical predictions and RHIC experimental results on heavy flavour production without including additional step related to the semileptonic decays. Recently, the formalism of the $k_t$-factorization approach has been found to give very good description of open charm \\cite{Maciula:2013wg} and bottom \\cite{Jung:2011yt,Karpishkov:2014epa} production rates and kinematical correlations in proton-proton scattering at $\\sqrt{s} = 7$ TeV measured by the ALICE, ATLAS, CMS and LHCb experiments. However, a significant sensitivity of theoretical predictions on the model of unintegrated (transverse momentum dependent) gluon distributions (UGDFs) used in calculations has been also reported.  \n\nTherefore, it is very interesting to make similar study for the STAR experimental data on open heavy flavour production at $\\sqrt{s} = 200$ and $500$ GeV. This         \nmay be a good test of different models of UGDFs in the RHIC kinematical regime where one can probe parton (gluon) distributons at intermediate longitudinal momentum fractions $x_1\/x_2 \\sim 10^{-2} - 10^{-1}$. Here, we wish to pay particular attention on UGDF models favoured by the LHC data and on a new up-to-date parametrizations based on the HERA collider DIS high-precision data. The present study is an important extension of our previous paper, where charm and bottom cross section at RHIC has been considered in the context of semileptonic decays of open heavy mesons \\cite{Luszczak:2008je}.\n \nPrecise predictions for charmed mesons may also shed new light on the theoretical understanding of non-photonic lepton production at RHIC.\nOur previous studies of these processes within the $k_t$-factorization approach were based on rather older models of UGDFs which may be the reason of the reported missing strength in description of the RHIC experimental data. Similar problem was also noticed within the NLO collinear calculations in the FONLL model, where only upper limits of the theoretical predictions are consistent with the relevant STAR and PHENIX data \\cite{Cacciari:2005rk}. Therefore, in the following paper we will also revise theoretical cross sections of the non-photonic lepton production at RHIC wihtin the $k_t$-factorizaton approach, taking as a reference point the results obtained in the analysis of the new hadron-level STAR data. In the following calculation, except of updated UGDFs, we also take into account the effect of transformation of semileptonic decay functions between laboratory ($e^{+}e^{-}$ center-of-mass system) and rest frames of decaying $D$ or $B$ mesons.\n\n\\section{Theoretical formalism}\n\nSeveral different mechanisms play a role in heavy quark hadroproduction. In general, in the framework of QCD there are two types of the $\\mathcal{O}(\\alpha_S^2)$ leading-order (LO) $2 \\to 2$ subprocesses: $q\\bar q \\to Q \\overline Q$ and $gg \\to Q \\overline Q$ \\cite{Combridge:1978kx}, often referred to as heavy quark-antiquark pair creation. The first mechanism, $q\\bar q$-annihilation, is important only near the threshold and at very large invariant masses of $Q\\overline Q$ system or extremely forward rapidities. This contribution is therefore especially important in the case of top quark production, however, for charm and bottom production, starting from RHIC, through Tevatron, up to the LHC it can be safely neglected. At high energies, production of charm and bottom flavours is dominated by the gluon-gluon fusion, which is the starting point of the following analysis. In the case of heavy quark production the $\\mathcal{O}(\\alpha_S^3)$ next-to-leading order (NLO) perturbative contributions have been found to be of special importance (see e.g. \\cite{Beenakker:1990maa}) and have to be included in order to describe the heavy flavour high energy experimental data.\n\nThe QED contributions, with one or two photons initiated reactions, such as $\\gamma g \\to Q \\overline Q$, $g \\gamma \\to Q \\overline Q$, $\\gamma \\gamma \\to Q \\overline Q$ have been carefully studied in Ref.~\\cite{Luszczak:2011uh} together with other sub-leading contributions to production of charm and were found to be negligibly small at high energies.\n\nIn the following, the cross sections for charm and bottom quark production in proton-proton collisions are calculated in the framework of the $k_t$-factorization approach. This framework has been successfully applied for different high energy processes, including heavy quark production (see e.g.~\\cite{Maciula:2013wg} and references therein). According to this approach, the transverse momenta $k_{t}$'s (virtualities) of partons which initiate reaction are taken into account and the sum of transverse momenta of the final $Q$ and $\\overline Q$ no longer\ncancels. Then the LO differential cross section for the $Q \\overline Q$ pair production reads:\n\\begin{eqnarray}\\label{LO_kt-factorization} \n\\frac{d \\sigma(h_1 h_2 \\to Q \\overline Q \\, X)}{d y_1 d y_2 d^2p_{1,t} d^2p_{2,t}} &=& \\sum_{i,j} \\;\n\\int \\frac{d^2 k_{1,t}}{\\pi} \\frac{d^2 k_{2,t}}{\\pi}\n\\frac{1}{16 \\pi^2 (x_1 x_2 s)^2} \\; \\overline{ | {\\cal M}^{off-shell}_{g^* g^* \\to Q \\overline Q} |^2}\n \\\\  \n&& \\times  \\; \\delta^{2} \\left( \\vec{k}_{1,t} + \\vec{k}_{2,t} \n                 - \\vec{p}_{1,t} - \\vec{p}_{2,t} \\right) \\;\n{\\cal F}_g(x_1,k_{1,t}^2) \\; {\\cal F}_g(x_2,k_{2,t}^2) \\; \\nonumber ,   \n\\end{eqnarray}\nwhere ${\\cal F}_g(x_1,k_{1,t}^2)$ and ${\\cal F}_g(x_2,k_{2,t}^2)$\nare the UGDFs for the both colliding hadrons. The extra integration is over transverse momenta of the initial\npartons. Explicit treatment of the transverse part of momenta makes the approach very efficient in studies of correlation observables. The two-dimensional Dirac delta function assures momentum conservation.\nThe unintegrated (transverse momentum dependent) gluon distributions must be evaluated at:\n\\begin{equation}\nx_1 = \\frac{m_{1,t}}{\\sqrt{s}}\\exp( y_1) \n     + \\frac{m_{2,t}}{\\sqrt{s}}\\exp( y_2), \\;\\;\\;\\;\\;\\;\nx_2 = \\frac{m_{1,t}}{\\sqrt{s}}\\exp(-y_1) \n     + \\frac{m_{2,t}}{\\sqrt{s}}\\exp(-y_2), \\nonumber\n\\end{equation}\nwhere $m_{i,t} = \\sqrt{p_{i,t}^2 + m_Q^2}$ is the quark\/antiquark transverse mass. In the case of heavy quark production at RHIC energies, especially in the central rapidity region, one test kinematical regime of $x > 10^{-2}$.  \n\nThe LO matrix element squared for off-shell gluons is taken in the analytic form proposed by Catani, Ciafaloni and Hautmann (CCH) \\cite{Catani:1990eg}. This analytic formula was basically derived within the standard QCD framework and can be adopted to the numerical analyses. It was also checked that the CCH expression is consistent with those presented later in Refs.~\\cite{Collins:1991ty,Ball:2001pq} and in the limit of $k_{1,t}^2 \\to 0$, $k_{2,t}^2 \\to 0$ it converges to the on-shell formula.\n\nThe calculation of higher-order corrections in the $k_t$-factorization is much more complicated than in the case of collinear approximation.\nHowever, the common statement is that actually in the $k_{t}$-factorization approach with LO off-shell matrix elements some part of higher-order corrections is effectively included. This is due to emission of extra gluons encoded\nin the unintegrated gluon densities. More details of the theoretical formalism adopted here can be found in Ref.\\cite{Maciula:2013wg}. \n  \nIn the numerical calculation below we have applied several unintegrated gluon densities which are based on different theoretical assumptions. The Kimber-Martin-Ryskin (KMR) UGDF is derived from a modified DGLAP-BFKL evolution equation \\cite{Kimber:2001sc,Watt:2003mx} and has been found recently to work very well in the case of charm and bottom production at the LHC. A special emphasis here is put on the UGDF models obtained as a solution of CCFM evolution equation. Here we use an older Jung set$B0$ parametrization \\cite{Jung:2004gs} and up-to-date JH2013 distributions \\cite{Hautmann:2013tba} determined from the fits to HERA high-precision DIS measurements. The JH2013 set1 is obtained from the fit to inclusive $F_{2}$ data only while JH2013 set2 is derived from the fit to both $F^{charm}_{2}$ and $F_{2}$ data. The UGDFs based on BFKL and BK equations are not applied in the following analysis since they are dedicated to smaller-$x$ values.    \n\nIn the calculation of charm and bottom quark cross sections the central value of numerical results is obtained with the renormalization and factorization scales $\\mu^2 = \\mu_{R}^{2} = \\mu_{F}^{2} = \\frac{m^{2}_{1,t} + m^{2}_{2,t}}{2}$ and quark mass $m_{c} = 1.5$ and $m_{b} = 4.75$ GeV for charm and bottom, respectively. The uncertainties of the predictions are estimated by changing quark mass by $\\pm 0.25$ GeV and by varying scales by a factor $2$.\nThe gray shaded bands drawn in the following figures represent these both sources of uncertainties summed in quadrature. The MSTW08LO \\cite{Martin:2009iq} collinear parton distribution function (PDF) is used for the calculation of the KMR unintegrated gluon density.\n\nThe transition from quarks and gluons to hadrons, called hadronization or parton fragmentation, can be so far approached only through phenomenological models. In principle, in the case of multi-particle final states the Lund string model \\cite{Andersson:1983ia} and the cluster fragmentation model \\cite{Webber:1983if} are often used. However, the hadronization of heavy quarks in non-Monte-Carlo calculations is usually done with the help of fragmentation functions (FF). The latter are similar objects as the parton distribution functions (PDFs) and provide the probability for finding a hadron produced from a high energy quark or gluon.\n\nConsidering fragmentation of a high energy quark (parton) $q$ with zero transverse momentum $p_t$ into a hadron $q \\to h + X$ one usually assumes that the transition is soft and does not add any transverse momentum. In consequence it is a delta function in transverse momentum.\n\nDefining $D(z)dz$ as the probability for the quark $q$ fragmenting into a hadron $h$ which carries a fraction $z$ of its longitudinal momentum, the spectrum of hadrons is given by\n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h d^2 p_{t,h}} = \\delta^{(2)}(\\vec{p}_{t,h}) \\int dz dx D(z) \\frac{d\\sigma_q}{dx}\\delta(x_h - zx) \\;\n\\end{equation}\nor \n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h} = \\int \\frac{dz}{z} D(z) \\frac{d\\sigma_q}{dx}\\Bigg\\vert_{x = x_h\/z} \\;.\n\\end{equation}\n\nThis can be generalized to the fragmentation at finite $p_t$ with intrinsic transverse momentum $\\kappa_t$. Thus the incoming quark has transverse momentum $p_{t,q}$ and the outgoing hadron transverse momentum can be decomposed as $p_{t,h} = \\kappa_t + z p_{t,q}$. Moving back to the situation where $ p_{t,q} = 0$ one can argue that the intrinsic transverse momentum $\\kappa_t$ must be small.\n\nIntroducing the transverse momentum dependent fragmentation probablity $D(z,\\kappa_t)dzd\\kappa_t$ one obtains\nthe hadron spectum as\n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int dz d^2\\kappa_t D(z,\\kappa_t) dx_q d^2 p_{t,q} \\frac{d\\sigma_q}{dx_q d^2 p_{t,q}}\n\\delta(x_h - zx_q) \\delta^{(2)}(\\vec{p}_{t,h} - \\vec{\\kappa_t} - z\\vec{p}_{t,q}) \\;.\n\\end{equation}\nHowever, one often neglects the intrinsic transverse momentum assuming:\n\\begin{equation}\nD(z,\\kappa_t) = D(z)\\delta^{(2)}(\\vec{\\kappa_t}).\n\\end{equation}\nThen the general formula reads\n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int dz d x_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dx_q d^2 p_{t,q}}\n\\delta(x_h - zx_q) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q}) \\;,\n\\end{equation}\nor integrating out the $\\delta$-functions\n\\begin{equation}\n\\frac{d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int \\frac{dz}{z^2} D(z) \\frac{d\\sigma_q}{dx_q d^2 p_{t,q}}\\Bigg\\vert_{x_q = x_h\/z \\atop p_{t,q} = p_{t,h}\/z } \\;.\n\\end{equation}\nIt is belived that this procedure provides correct implementation of small intrinsic transverse momentum into the splitting. Since the hadron on-shell four momentum is fully specified by $x_h$ and $p_{t,h}$, starting from the above formula one can calculate many different distributions. Especially conversion to the rapidity distributions is very simple because of the trivial jacobian:\n\\begin{equation}\n\\frac{xd\\sigma}{dx d^2 p_t} = \\frac{d\\sigma}{dy d^2 p_t}.\n\\end{equation}\nIt can be further written\n\\begin{eqnarray}\n\\frac{d\\sigma_h}{dy_h d^2 p_{t,h}}\\!&=&\\!\\frac{x_h d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int dz dx_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dx_q d^2 p_{t,q}}\nx_h \\cdot \\delta(x_h - z x_q) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q})\\nonumber \\\\\n&=& \\int dz \\frac{dx_q}{x_q} d^2 p_{t,q} D(z) \\frac{x_q d\\sigma_q}{dx_q d^2 p_{t,q}}\nx_h \\cdot \\delta(x_h - z x_q) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q}).\n\\end{eqnarray}\nNeglecting masses, one has\n\\begin{equation}\nx_h = \\frac{p_{t,h}}{\\sqrt{s}} e^{y_h}, \\;\\;\\;\\;\\;\\; x_q = \\frac{p_{t,q}}{\\sqrt{s}} e^{y_q} \\;,\n\\end{equation}\nso that,\n\\begin{eqnarray}\n\\frac{d\\sigma_h}{dy_h d^2 p_{t,h}}\\!&=&\\!\\frac{x_h d\\sigma_h}{dx_h d^2 p_{t,h}} = \\int dz dy_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}}\n\\delta(1 - e^{(y_q - y_h)}) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q})\\nonumber \\\\\n&=& \\int dz dy_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}}\n\\delta(y_q - y_h) \\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q}) \\nonumber \\\\\n&=& \\int \\frac{dz}{z^2} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}} \\Bigg\\vert_{y_q = y_h \\atop p_{t,q} = p_{t,h}\/z} \\;.\n\\end{eqnarray}\nThis way one gets (reproduces) the standard formula for the fragmentation in the case of light hadrons.\n\nWhen going to more general case, taking masses into account and introducing $m_t = \\sqrt{p_t^2 + m^2}$, one gets\n\\begin{equation}\nx_h = \\frac{m_t^h}{\\sqrt{s}} e^{y_h}, \\;\\;\\;\\;\\;\\; x_q = \\frac{m_t^q}{\\sqrt{s}} e^{y_q} \\;,\n\\end{equation}\nor\n\\begin{equation}\ny_h = \\log \\left( \\frac{x_h\\sqrt{s}}{m_t^h} \\right), \\;\\;\\;\\;\\;\\; y_q = \\log \\left( \\frac{x_q\\sqrt{s}}{m_t^q} \\right), \n\\end{equation}\nthen, the $z$-dependent rapidity shift between quark and hadron reads\n\\begin{equation}\n\\delta y = y_q - y_h = \\log \\left( \\frac{x_q m_t^h}{x_h m_t^q} \\right) = \\log \\left( \\frac{m_t^h}{z m_t^q} \\right).\n\\end{equation}\nThe delta function now becomes\n\\begin{eqnarray}\nx_h \\delta(x_h - x_q z) &=& \\delta \\left( 1 - \\frac{z x_q}{x_h} \\right)= \\delta \\left( 1- \\frac{z m_t^q}{m_t^h} e^{(y_q - y_h)} \\right) \\nonumber \\\\\n&=&\n\\delta \\left( 1 - e^{(y^q - \\delta y - y_h)} \\right) = \\delta(y_q - \\delta y - y^h).\n\\end{eqnarray}\nFinally,\n\\begin{eqnarray}\n\\frac{d\\sigma_h}{dy_h d^2 p_{t,h}} &=& \\int dz dy_q d^2 p_{t,q} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}}\n\\delta(y_h - y_q + \\delta y)\\delta^{(2)}(\\vec{p}_{t,h} - z \\vec{p}_{t,q}) \\nonumber \\\\\n&=& \\int \\frac{dz}{z^2} D(z) \\frac{d\\sigma_q}{dy_q d^2 p_{t,q}}\\Bigg\\vert_{y_q = y_h + \\delta y \\atop p_{t,q} = p_{t,h}\/z} \\;. \n\\end{eqnarray}\nIn turn, using $p_{t,q} = p_{t,h}\/z$, the rapidity shift $\\delta y$ can be rewritten\n\\begin{equation}\n\\delta y = \\frac{1}{2}\\log \\left( \\frac{p^2_{t,h} + m_h^2}{p^2_{t,h} + z^2 m_q^2} \\right).\n\\end{equation}\n\nIt is clear that the rescalling of the transverse momentum is the most important effect. This is because one deals with very steep functions of transverse momenta. From the reason that rapidity spectra are usually flat, or slowly varying, the shift $\\delta y$ is not so important. In fact, it is entirely negligible, if $p^2_{t,h} \\gg m_q^2, m_h^2$. The shift is most important at very small $p^2_{t,h} \\ll m_q^2, m_h^2$, where it becomes\n\\begin{equation}\n\\delta y \\sim \\log\\left( \\frac{m_h}{zm_q}  \\right) \\approx \\log\\left( \\frac{1}{z} \\right). \n\\end{equation}\nIt is worth to notice, that at finite $p_{t,h}$ it should never be really large: small $z$ is damped by the fact that the quark spectrum drops rapidly as a function of $p_{t,q} = p_{t,h}\/z$. However, at $p_{t,h}$ = 0, that suppression causes an effect and the whole integral over $z$ becomes important, with very small $z$ causing large rapidity shifts. Fortunately, for heavy quarks, the fragmentation function is peaked at large $z$ (see e.g. \\cite{Luszczak:2014cxa}). Moreover, one has to remember, that taking into account the small-$z$ region in the fragmentation function is theoretically not warranted, since the standard DGLAP approach to fragmentation breaks down in this region. \n\nTaking all together, according to the above formalism, in the following numerical calculations\nthe inclusive distributions of open charm and bottom hadrons $h =D, B$ are obtained through a convolution of inclusive distributions of heavy quarks\/antiquarks and $Q \\to h$ fragmentation functions:\n\\begin{equation}\n\\frac{d \\sigma(pp \\rightarrow h \\bar{h} X)}{d y_h d^2 p_{t,h}} \\approx\n\\int_0^1 \\frac{dz}{z^2} D_{Q \\to h}(z)\n\\frac{d \\sigma(pp \\rightarrow Q \\overline{Q} X)}{d y_Q d^2 p_{t,Q}}\n\\Bigg\\vert_{y_Q = y_h \\atop p_{t,Q} = p_{t,h}\/z} \\;,\n\\label{Q_to_h}\n\\end{equation}\nwhere $p_{t,Q} = \\frac{p_{t,h}}{z}$ and $z$ is the fraction of\nlongitudinal momentum of heavy quark $Q$ carried by a hadron $h$.\nThe origin why the approximation typical for light hadrons assuming that $y_{Q}$ is\nunchanged in the fragmentation process, i.e. $y_h = y_Q$, is also applied in the case of heavy hadrons was carefully clarified in the previous paragraph and is commonly accepted. \n\nAs a default set in all the following numerical calculations the standard Peterson model of fragmentation function \\cite{Peterson:1982ak} with the parameters $\\varepsilon_{c} = 0.02$ and $\\varepsilon_{b} = 0.001$ is applied. This choice of fragmentation function and parameters is based on our previous theoretical studies of open charm production at the LHC \\cite{Maciula:2013wg}, where detailed analysis of uncertainties related with application of different models of FFs was done. Here, we decided not to repeat all the previously analyzed issues and take into consideration only the most data-favoured scenario\\footnote{This is also consistent with prescription applied in the FONNL framework, where rather harder fragmentation functions are suggested \\cite{Cacciari:2005rk}.}. The main conclusions should not change when moving from LHC to RHIC energies and the uncertainties due to the fragmentation effects may be neglected with respect to those related to the perturbative part of the calculation. \n  \nIn the calculations of the cross sections for $D^{0}$ and $D^{*}$ mesons the fragmentation functions should be normalized to the relevant branching fractions $\\textrm{BR}(c \\to D)$, e.g. from Ref.~\\cite{Lohrmann:2011np}. However, the measured by STAR differential distributions for $D^{0}$ and $D^{*}$ meson are normalized to the parton-level $c\\bar c$ cross section which simply means that the $\\textrm{BR}(c \\to D) = 1$ should be taken in numerical calculations.\n\nTheoretical predictions for production of non-photonic leptons in proton-proton scattering is a three-step process. The whole procedure can be written in the following schematic way:\n\\begin{equation}\n\\frac{d \\sigma(pp \\to l^{\\pm} X )}{d y_e d^2 p_{t,e}} =\n\\frac{d \\sigma(pp \\to Q X)}{d y_Q d^2 p_{t,Q}} \\otimes\nD_{Q \\to h} \\otimes\nf_{h \\to l^{\\pm}} \\; ,\n\\label{whole_procedure}\n\\end{equation}\nwhere the symbol $\\otimes$ denotes a generic convolution. Thus, the cross section for non-photonic leptons is a convolution of the cross section for heavy quarks with fragmentation function $D_{Q \\to h}$ and with semileptonic decay function\n$f_{h \\to l^{\\pm}}$ for heavy mesons.\n\nIn principle, the semileptonic decays can be calculated \\cite{Artuso:2008vf}.\nThe simplest approach to describe the decays of $D$ and $B$ mesons is given by the spectator model \\cite{Artuso:2008vf}, where the QCD effects from the higher-order corrections between heavy $Q$ and light $q$ quarks are neglected. This model works better for bottom quarks since there the mass sufficiently suppresses these corrections. In the case of charm, the QCD effects become more important but they can be also qualitatively modelled.\n\nSince there are many decay channels with different\nnumber of particles the above procedure is not easy and rather labor-intensive. It introduces some model \nuncertainties and requires inclusion of all final state channels explicitly.\n\nAn alternative way to incorporate semileptonic decays into theoretical model is to take relevant experimental input. For example, the CLEO \\cite{Adam:2006nu} and BABAR \\cite{Aubert:2004td} collaborations have measured very precisely the momentum spectrum of electrons\/positrons coming from the decays of $D$ and $B$ mesons, respectively. This is done by producing resonances: $\\Psi(3770)$\nwhich decays into $D$ and $\\bar D$ mesons, and $\\Upsilon(4S)$ which decays into $B$ and $\\bar B$ mesons.\n\nThis less ambitious but more pragmatic approach is based on purely empirical fits to (not absolutely normalized) CLEO and BABAR experimental data points. These electron decay functions should account for the proper\nbranching fractions which are known experimentally (see e.g. \\cite{Beringer:1900zz,Adam:2006nu,Aubert:2004td}).\nThe branching fractions for various species of $D$ mesons are different:\n\\begin{eqnarray}\n&&\\mathrm{BR}(D^+\\to~e^+ \\nu_e X)=16.13\\pm 0.20(\\mathrm{stat.})\\pm 0.33(\\mathrm{syst.})\\%,\\nonumber \\\\\n&&\\mathrm{BR}(D^0\\to~e^+ \\nu_e X)=6.46\\pm 0.17(\\mathrm{stat.})\\pm 0.13(\\mathrm{syst.})\\%. \n\\end{eqnarray}\nBecause the shapes of positron spectra for both decays are \nidentical within error bars we can take the average value of BR($D\\!\\to\\!e \\, \\nu_e \\, X) \\approx 10 \\%$ and \nsimplify the calculation. In turn, the branching fraction of open bottom is found to be:\n\\begin{equation}\n\\mathrm{BR}(B\\to e \\, \\nu_e \\, X) = 10.36 \\pm 0.06(\\mathrm{stat.}) \\pm 0.23(\\mathrm{syst.}) \\% .\n\\end{equation} \n\nAfter renormalizing to experimental branching fractions the adjusted decay functions are then use to generate leptons \nin the rest frame of the decaying $D$ and $B$ mesons in a Monte Carlo approach. This way one can avoid all uncertainties associated with explicit calculations of semileptonic decays of mesons.\n\nIn both cases the heavy mesons are almost at rest, so in practice one measures the meson rest frame\ndistributions of electrons\/positrons. With this assumption one can find a good fit to the CLEO and BABAR data with:\n\\begin{eqnarray}\nf^{Lab}_{CLEO}(p) &=& 12.55 (p+0.02)^{2.55} (0.98-p)^{2.75}\\;, \\label{CLEO_fit_function} \\\\\nf^{Lab}_{BABAR}(p) &=& \\left( 126.16+14293.09 \\exp(-2.24 \\log(2.51-0.97 p)^2 \\right) \\nonumber \\\\\n&&\\times \\left( -41.79+42.78 \\exp(-0.5(|p-1.27|) \/1.8 )^{8.78} \\right )\\;.\n \\label{BABAR_fit_function}\n\\end{eqnarray}\nIn these purely empirical parametrizations $p$ must be taken in GeV.\n\nIn order to take into account the small effect of the non-zero motion of the $D$ mesons in \nthe case of the CLEO experiment and of the $B$ mesons in the case of the BABAR experiment, the above parametrizations of the fits in the laboratory frames have to be modified. The improvement can be achieved by including the boost of the new modified rest frame functions to the CLEO and BABAR laboratory frames. The quality of fits from Eqs.~(\\ref{CLEO_fit_function}) and (\\ref{BABAR_fit_function}) will be reproduced. The $D$ and $B$ rest frame decay functions take the following form:\n\\begin{eqnarray}\nf^{Rest}_{CLEO}(p) &=& 12.7 (p+0.047)^{2.72} (0.9-p)^{2.21}\\;, \\label{CLEO_fit_boost}  \\\\\nf^{Rest}_{BABAR}(p) &=& \\left( 126.16+14511.2 \\exp(-1.93 \\log(2.7-1.0825 p)^2 \\right) \\nonumber \\\\\n&&\\times \\left( -41.79+42.78 \\exp(-0.5(|p-1.27|) \/1.8 )^{8.78} \\right )\\;.\n \\label{BABAR_fit_boost}\n\\end{eqnarray}\n\n\\begin{figure}[tb]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{CLEO_decay.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{BABAR_decay.eps}}\n\\end{minipage}\n   \\caption{\n\\small Fits to the CLEO (left) and BABAR (right) data. The solid lines correspond to the parametrizations in the laboratory frames and the dashed lines to the meson rest frames, which represent incorporation of effects related to the non-zero motion of decaying mesons.}\n \\label{fig:p-decay-1}\n\\end{figure}\n\nBoth, laboratory and rest frame parametrizations of the semileptonic decay functions for $D$ and $B$ mesons are drawn in Fig.~\\ref{fig:p-decay-1} together with the CLEO (left panel) and BABAR (right panel) experimental data. Some small differences between the different parametrizations appear only at larger values of electron momentum. The influence of this effect on differential cross sections of non-photonic leptons is expected to be negligible and will be shown when presenting numerical results. Our analytical formulas for the rest frame decay functions only slightly differ from those obtained in Ref.~\\cite{Bolzoni:2012kx}.\n\nThe theoretical model for non-photonic lepton production in hadronic reactions described here has been recently found to give a very good description of the experimental data collected with the ALICE detector at the LHC \\cite{Abelev:2014hla,Abelev:2014gla}.\n\n\\section{Numerical results}\n\nThe total cross section for charm production in $pp$ scattering extracted from the STAR measurement of $D^{0}$ and $D^{*}$ mesons at $\\sqrt{s} = 200$ GeV is $\\sigma_{c\\bar{c}} = 797\\pm210^{208}_{295}$ $\\mu$b. Corresponding calculated total cross sections is $\\sigma_{c\\bar{c}}^{\\mathrm{setB}0} = 541$ with the CCFM Jung set$B0$ UGDF. The calculated value is consistent with the measured value taking into account large experimental uncertainties. The total cross section at $\\sqrt{s} = 500$ GeV using the same UGDF is predicted to be $\\sigma_{c\\bar{c}}^{\\mathrm{setB}0} = 1006$ $\\mu$b.\n\nThe STAR collaboration also carried out a measurement of charm production cross sections at midrapidity $\\frac{d\\sigma_{c\\bar{c}}}{dy}|_{y = 0}$. Comparison of the experimental results and the theoretical ones is presented in Table \\ref{tableRHIC-Dmesons}.  \nHere again the CCFM Jung set$B0$ UGDF give results consistent with the measurements.\n\n\\begin{table}[tb]%\n\\caption{The midrapidity $\\frac{d\\sigma_{c\\bar{c}}}{dy}|_{y = 0}$ cross section for charm production in proton-proton scattering at $\\sqrt{s} = 200$ and $500$ GeV: the STAR results versus results of calculation with the Jung set$B0$ UGDF.}\n\\newcolumntype{Z}{>{\\centering\\arraybackslash}X}\n\\label{tableRHIC-Dmesons}\n\\centering %\n\\begin{tabularx}{1.0\\textwidth}{ZZZZ}\n\\toprule[0.1em] %\n\\\\[-3.4ex] \n\\multicolumn{2}{c}{Experiment: STAR, $\\frac{d\\sigma_{c\\bar{c}}}{dy}|_{y = 0}$}  & Theory: Jung set$B0$ \\\\ [+0.4ex]\n   \n\\bottomrule[0.1em]\n     & \\\\ [-3.0ex]\n $\\sqrt{s} = 200$ GeV  &  $170\\pm45^{+38}_{-59}$ $\\mu$b  & $130$ $\\mu$b \\\\ [+0.4ex]\n $\\sqrt{s} = 500$ GeV  &  $217\\pm86\\pm73$ $\\mu$b         & $191$ $\\mu$b \\\\ [+1.4ex]\n\n\\bottomrule[0.1em]\n\n\\end{tabularx}\n\n\\end{table}\n\n\\subsection{Open charm mesons}\n\nFigure \\ref{fig:pt-star-D1} presents transverse momentum distributions of charmed mesons in proton-proton collisions at $\\sqrt{s} = 200$ GeV for $|y_{D}| < 0.5$ (left panel) and at $\\sqrt{s} = 500$ GeV for $|y_{D}| < 1.0$ (right panel) together with the STAR data points. Both experimental data and theoretical results are normalized to the $c\\bar c$ parton-level cross section dividing by the $c \\to D^*, D^0$ fragmentation fractions. Results of numerical calculations obtained with the KMR (dotted line), the JH2013 set1 (long-dashed-dotted line), set2 (long-dashed line) and the Jung setB$0$ (solid line) UGDFs are shown. At both energies very good description of the experimental data is obtained with the Jung setB$0$ UGDF. Results calculated with the JH2013 set1 UGDF overestimate the data points in the whole range of measured $p_t$'s.\nThe JH2013 set2 and the KMR UGDFs significantly underestimate the distribution measured at $\\sqrt{s} = 200$ GeV and the situation is only slightly improved in the case of the higher energy where the results of both of them reach the two last data points at larger transverse momenta.    \n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dpt_STAR_D_updfs_eps02_200.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dpt_STAR_D_updfs_eps02_500.eps}}\n\\end{minipage}\n   \\caption{\n\\small The transverse momentum distribution of $D$ mesons normalized to the parton-level cross section at $\\sqrt{s} = 200$ (left) and $500$ GeV (right). The STAR experimental data points are compared to the results of the $k_{t}$-factorization calculations with the KMR (dotted line), the JH2013 set1 (long-dashed-dotted line), set2 (long-dashed line) and the Jung setB$0$ (solid line) UGDFs. }\n \\label{fig:pt-star-D1}\n\\end{figure}\n\nMain uncertainties of the theoretical calculations coming from the perturbative part are shown in Fig.~\\ref{fig:pt-D-star-uncert}.\nThe shaded bands represent uncertainties of the calculations with the Jung setB$0$ UGDF related to the choice of the factorization and\/or renormalization scales and those due to the charm quark mass. The result from the FONLL approach is also drawn for comparison. The uncertainties are larger at lower transverse momenta, where the effects of quark mass uncertainties are more important, and decrease with increasing $p_t$. The FONLL predictions underestimate the experimental data almost in the whole measured range. Their central value coincides with the lower-limit of the $k_{t}$-factorization predictions with the Jung setB$0$ UGDF.\n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dpt_STAR_D_setB0_uncert_eps02_200.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dpt_STAR_D_setB0_uncert_eps02_500.eps}}\n\\end{minipage}\n   \\caption{\n\\small The uncertainties of the theoretical predictions for Jung setB$0$ UGDF at $\\sqrt{s} = 200$ (left) and $500$ GeV (right). Uncertainties due to the choice of the factorization and\/or renormalization scales and those related to the charm quarks mass are summed in quadrature. In the case of $\\sqrt{s} = 200$ GeV data the results obtained within the FONLL framework are drawn for comparison. Details are specified in the plots.}\n \\label{fig:pt-D-star-uncert}\n\\end{figure}\n\n\n\\subsection{Non-photonic electrons}\n\nA first theoretical investigation of the non-photonic electron production at RHIC within the framework of the $k_t$-factorization was performed in Ref.~\\cite{Luszczak:2008je}. Some missing strength in the description of the measured differential distributions has been reported there, especially in the region of small transverse momenta. In the meantime, the STAR collaboration has published new measurements of non-photonic electrons with separated charm and bottom contributions \\cite{Agakishiev:2011mr}. Therefore, it is very interesting to make a revision of theoretical cross sections taking into account the new\nresults from the hadron-level analysis of charm production at RHIC. Here, our previous results from Ref.~\\cite{Luszczak:2008je} are updated by the application of the Jung setB$0$ UGDF which as was shown in the previous subsection works very well for the STAR data on open charm meson production.\n\nThe experimental cross sections for charm and bottom flavoured electrons measured at RHIC are collected in Table \\ref{table-NPE-RHIC}. The values calculated with the Jung setB$0$ UGDF are consistent with the measurements.\n\n\\begin{table}[!h]\\small%\n\\caption{The experimental and theoretical cross sections $\\frac{d\\sigma}{dy_{e}}|_{y_{e} = 0}$ for non-photonic electron production in proton-proton scattering at $\\sqrt{s} = 200$ GeV.}\n\\newcolumntype{Z}{>{\\centering\\arraybackslash}X}\n\\label{table-NPE-RHIC}\n\\centering %\n\\begin{tabularx}{1.0\\textwidth}{ZZZZ}\n\\toprule[0.1em] %\n\\\\[-2.4ex] \n                  \\multicolumn{2}{c}{Experiment}  & Theory,  Jung setB0 \\\\ [+0.4ex]\n                  \n\\bottomrule[0.1em]\n     & \\\\ [-2.0ex]\n STAR, $pp\\rightarrow c\\bar{c} X \\rightarrow e X'$                               &  \\multirow{2}*{$6.2\\pm0.7\\pm1.5$ nb}         & \\multirow{2}*{$7.55$ nb } \\\\ [+0.4ex]\n        $\\frac{d\\sigma}{dy_{e}}|_{y_{e} = 0}$,  $3<p_{\\perp}<10$ GeV          &                                                 &       \\\\ [+1.4ex]\n        \n  STAR,  $pp\\rightarrow b\\bar{b} X \\rightarrow e X'$                               &  \\multirow{2}*{$4.0\\pm0.5\\pm1.1$ nb}   & \\multirow{2}*{$6.65$ nb}    \\\\ [+0.4ex]       \n   $\\frac{d\\sigma}{dy_{e}}|_{y_{e} = 0}$,  $3<p_{\\perp}<10$ GeV                  &                                     &       \\\\ [+1.4ex] \n\n PHENIX, $pp\\rightarrow c\\bar{c} X \\rightarrow e X'$                               &  \\multirow{2}*{$5.95\\pm0.59\\pm2.0$ $\\mu$b}         & \\multirow{2}*{$5.09$ $\\mu$b } \\\\ [+0.4ex]\n        $\\frac{d\\sigma}{dy_{e}}|_{y_{e} = 0}$,  $p_{\\perp}>0.4$ GeV          &                                                 &       \\\\ [+1.4ex]\n\n\\bottomrule[0.1em]\n\n\\end{tabularx}\n\n\\end{table}\n\nFigure~\\ref{fig:pt-rhic-npe-1} shows the transverse momentum distributions of electrons from semileptonic decays of charm flavoured hadrons $H_c$ (left panel) and from bottom hadrons $H_b$ (right panel) measured by STAR. The experimental data is compared to the numerical results for the Jung setB$0$ UGDF. The theoretical uncertainties coming from the perturbative part of calculations are also shown for completeness. The rest frame semileptonic decay functions from Eqs.~(\\ref{CLEO_fit_boost}) and (\\ref{BABAR_fit_boost}) are used. It is also assumed that the charm and bottom baryons decay semileptonically in the same way as $D$ and $B$ mesons, and therefore baryonic contributions may be effectively included by treating the baryons as mesons and taking BR$(c \\to D$; $b \\to B) = 1$ . The numerical results very well descibe the experimental data. The central value of the Jung setB$0$ UGDF give distributions that are sligthly above the predictions of the FONLL central value, especially in the small-$p_t$ region. In this case also the JH2013 set1 UGDF reasonably describes the data points taking into account experimental uncertainties.\nAs in the case of open charm data, the lines that corespond to the KMR and JH2013 set2 UGDFs lie much below the measured lepton distributions for both, charm and bottom components.    \n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_charm.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_bottom.eps}}\n\\end{minipage}\n   \\caption{\n\\small Transverse momentum distributions for electrons from semileptonic decays of charm (left) and bottom hadrons (right) measured in $pp$ scattering at $\\sqrt{s} = 200$ GeV. The STAR experimental data are compared to the $k_t$-factorization theoretical predictions obtained with different UGDFs as well as to the FONLL results. Theoretical uncertainties due to quark mass and scales variation are also shown. Further details are specified in the figures. }\n \\label{fig:pt-rhic-npe-1}\n\\end{figure}\n\n\n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_sum.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_sum_uncert.eps}}\n\\end{minipage}\n   \\caption{\n\\small Transverse momentum distributions of electrons coming from both charm and bottom hadrons summed together $H_{c+b}$, measured in $pp$-scattering at $\\sqrt{s} = 200$ GeV. The STAR and PHENIX experimental data are compared to the theoretical predictions obtained with the Jung setB$0$ UGDF. Separated charm and bottom contributions (left) and theoretical uncertainties due to quark mass and scales variation (right) are also shown. The FONLL predictions are drawn for comparison. Further details are specified in the figures. }\n \\label{fig:pt-rhic-npe-2}\n\\end{figure}\n\nIn Fig.~\\ref{fig:pt-rhic-npe-2} the results for summed contributions of charm and bottom flavours are shown.\nHere, the results of calculations are compared to the experimental distributions of heavy flavour electrons, that contain both charm and bottom components.\nThe left panel presents separately charm and bottom contributions as well as the sum of them. In the right panel, the uncertainties of the predictions for the Jung setB$0$ UGDF are drawn together with the lines corresponding to the FONLL results.\nIn contrast to the previous studies in Ref.~\\cite{Luszczak:2008je}, here the $k_t$-factorization results give excellent description of the STAR and PHENIX data, sligthly better than those from the FONLL approach, which are almost identical to the lower for the Jung setB$0$ UGDF. The crossing point between charm and bottom components is found to lies roughly at $p_t = 4$ GeV, which is in agreement with other theoretical investigations (see e.g. Ref.~\\cite{Cacciari:2005rk}).  \n\n\n\\begin{figure}[!h]\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_charm_fdecay.eps}}\n\\end{minipage}\n\\hspace{0.5cm}\n\\begin{minipage}{0.47\\textwidth}\n \\centerline{\\includegraphics[width=1.0\\textwidth]{dsig_dptl_RHIC_200_elec_bottom_boost.eps}}\n\\end{minipage}\n   \\caption{\n\\small The effect of using two different sets of the semileptonic decays functions, discussed in the present paper, at $\\sqrt{s} = 200$ GeV. The results obtained with the laboratory frame fits from Eqs.~(\\ref{CLEO_fit_function}) and (\\ref{BABAR_fit_function}) are compared to the meson rest frame fits from Eqs.~(\\ref{CLEO_fit_boost}) and~(\\ref{BABAR_fit_boost}). }\n \\label{fig:pt-rhic-npe-ratio-1}\n\\end{figure}\n\nFinally, the effects related to the fitting procedure of the CLEO and BABAR semileptonic data are depicted in Fig.~\\ref{fig:pt-rhic-npe-ratio-1}.\nHere, both sets of the semileptonic decay functions are used. As can be observed from the figure these effects do not really affect the electron spectra at RHIC. The difference is very small and sligthtly increases at higher transverse momenta. In the case of charm flavour (left panel), the application of the boosted decay function leads to a damping of the cross section by about $20\\%$ at $p_t = 10$ GeV. For the bottom flavour (right panel) the corresponding suppresion is only about $5\\%$, which is completely negligible.  \n\n\n\\section{Conclusions}\n\nIn this paper we have dicussed production of charm mesons and non-photonic\nelectrons in proton-proton scattering at the BNL RHIC.\nThe calculation of the charm quark-antiquark pairs has been performed\nin the framework of $k_t$-factorization approach which effectively\nincludes higher-order pQCD corrections.\n\nWe have used different models of unintegrated gluon distributions\nfrom the literature, including those that were applied recently to describe\ncharm data at the LHC and others used to descibe HERA deep-inelastic scattering data.\nThe hadronization of heavy quarks to mesons has been done by means of fragmentation\nfunction technique. The theoretical transverse momentum distributions of charmed\nmesons has been compared with recent experimental data of the STAR collaboration\ncollected at $\\sqrt{s}$ = 200 and 500 GeV.\nWe have carefully quantified uncertainties related to the choice of factorization\/renormalization\nscales as well as quark\/antiquark masses.\nWe have obtained very good agreement with the measured cross sections\nfor the Jung setB$0$ UGDF.\nFurthermore, our results have been compared with the results of the FONLL model.\nThe two approaches give rather similar results.\n\nSemileptonic decays of charmed and bottom mesons have been included\nvia empirical decay functions fitted to the CLEO and BABAR ($e^+ e^-$) data for vector meson decays.\nWe have shown that the inclusion of kinematical boost from meson ($D$ or $B$) \nrest frame to the $e^+ e^-$ center of mass (laboratory) system leads to only \nsmall modifications of the resulting decay functions and as a consequence \nalso for the distributions of non-photonic electrons in proton-proton collisions \nat the RHIC energies.\nConsequently we have obtained a rather good description of the electron\/positron \ntransverse momentum distributions of the STAR collaboration with the same UGDF as \nfor the charmed mesons.\nThis also demonstrates indirectly consistency of the meson and \nnon-photonic electron data.\n\n\n\\vspace{1cm}\n\n{\\bf Acknowledgments}\n\nWe are particularly indebted to Jaro Bielcik for a discussion of experimental aspects and to Wolfgang Sch{\\\"a}fer for a detailed discussion of effects related to quark-to-meson transition. This study was partially supported by the Polish National Science Centre grant DEC-2013\/09\/D\/ST2\/03724.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe hard--exclusive photo electroproduction off a nucleon, among a\nclass of hard exclusive but inelastic processes, is generally\nconsidered as the theoretic cleanest process to gain access to the\nso--called generalized parton distributions (GPDs)\n\\cite{MueRobGeyDitHor94,Ji96}. These non--perturbative\ndistributions contain manifold information that cannot be gained\nfrom measurements of deep (semi--)inelastic or elastic and other\nexclusive processes \\cite{BroLep80}. Based on a partonic\ninterpretation, GPDs allow us to reveal the internal structure of\nthe probed hadron, especially of the nucleon, from a new\nperspective, for comprehensive reviews see Ref.\\\n\\cite{Die03aBelRad05}.\n\nWe recall that the process in question has two interfering\nsubprocesses, namely, the Bethe--Heitler brems\\-strahlung and\ndeeply virtual Compton scattering (DVCS). The former is known in\nterms of the electromagnetic form factors and so the latter can be\nmeasured in several asymmetries, see e.g., Ref.\\ \\cite{DVCS}, that\nappear in leading order of the expansion with respect to the\ninverse photon virtuality, i.e., single and double spin and charge\nasymmetries\n\\cite{DieGouPirRal97,BelMueNieSch00,BelMueKir01,KirMue03}. Relying\non the validity of the operator product expansion (OPE) of two\nelectromagnetic currents, the DVCS amplitude factorizes to leading\npower, i.e., twist--two, in short-- and long--distance physics, where\nthe former and latter are incorporated in (resummed)\nWilson--coefficients and GPDs, respectively\n\\cite{MueRobGeyDitHor94}. Equivalently, in the partonic framework,\nthe DVCS amplitude to leading power accuracy is represented as\nconvolution of hard--scattering amplitudes, systematically\ncalculable in powers of the strong coupling, and GPDs. The\nfactorization of collinear singularities was shown to all orders\nin perturbation theory \\cite{ColFre98}.\n\nLet us remind that for experimental accessible scales, the\nperturbative description of elastic exclusive processes is\ncontroversially debated \\cite{IsgLle89a}. With respect to this\nfact one might wonder whether the perturbative GPD framework is\njustified. For the GPD phenomenology it is essential to clarify\nthis issue; unfortunately, this is an interlaced task.  The\nrelative size of perturbative and non--perturbative, i.e., power\nsuppressed contributions,  might serve as criteria for the\nreliability of the perturbative framework. The DVCS\nhard--scattering amplitudes (Wilson--coefficients) were\nperturbatively evaluated up to next--to--leading order (NLO) in the\nstrong coupling \\cite{BelMue97a,ManPilSteVanWei98}. It turned out\nthat these radiative corrections  can be of the order of 30\\% to\n50\\% for fixed target kinematics \\cite{BelMueNieSch99}. Even much\nlarger corrections have been reported for the kinematics of\ncollider experiments \\cite{FreMcD01b}. Mainly, but not only, this\nis due the appearances of gluonic GPDs at NLO, see also discussion\nin Sect.\\ 6.2.3 of Ref.\\ \\cite{BelMueKir01}. The problem arises\nwhether these sizeable corrections are related to the appearance\nof gluons, are induced by an awkward choice of GPD ans\\\"atze, or\nindicate that at experimental accessible scales the perturbative\nregime has not be fully set in.\n\nTo get a deeper insight into this issue,  we study in this letter\nperturbative corrections to the DVCS process beyond NLO accuracy\nwithin a framework that avoids a cumbersome diagrammatic\ncalculation. Based on the conformally covariant OPE (COPE),  the\nperturbative corrections up to  next--to--next--to--leading order\n(NNLO) can be economically evaluated  \\cite{Mue97a,BelMue97a}, see\nalso  review \\cite{BraKorMue03}. Thereby we restrict ourselves to\nthe flavor non--singlet part of the helicity conserved twist--two\namplitude in the parity even sector, which is the dominant\ncontribution in several DVCS observables. In Sect.\\\n\\ref{Sec-GenFor} we outline the  evaluation of this amplitude by\nmeans of the COPE \\cite{Mue97a,BelMue97a,MelMuePas02}. This yields\na divergent series of conformal GPD moments, which is resummed\nwithin a Mellin--Barnes integral. In Sect.\\ \\ref{Sec-NumRes} we\nnumerically study the radiative corrections, mainly due to the\nWilson--coefficients, at NLO and NNLO. Especially, we explore the\nnumerical differences of the NLO corrections in the modified\nminimal subtraction ($\\overline{\\rm MS}$) and conformal\nsubtraction (CS) schemes. We then provide  estimates to NNLO\naccuracy. Finally, in Sect.\\ \\ref{Sec-Con} we conclude.\n\n\n\\section{General formalism}\n\\label{Sec-GenFor}\n\nLet us first recall the standard perturbative QCD framework.\nUsually, one employs the $\\overline{\\rm MS}$ scheme, which is\nbased on dimensional regularization and the removal of the poles\nwith respect to the dimensional parameter $(4-n)\/2$. This scheme\nis used twofold: (i) to define the renormalized strong coupling\nand (ii) for the factorization of collinear singularities or if\none wishes  for the renormalization of (leading twist) composite\noperators\\footnote{To leading twist approximation there is an\none--to--one cross talk between the ultra--violet and collinear\nsingularities in the partonic matrix elements of these\noperators.}, which are labelled by the good quantum number spin.\nIn exclusive reactions such operators with given spin are also be\nbuild within total derivatives. In general operators with the same\nspin will mix under renormalization. To get rid of this mixing\nphenomenon, one changes at some stage of the full calculation in an\nexplicit or implicit manner to a basis of multiplicatively\nrenormalizable operators, i.e., one diagonalizes the evolution\nequation \\cite{EfrRad80}. For vanishing $\\beta(\\alpha_s)$ function\nthis  transformation leads to conformal\noperators \\cite{Mue94}.  Such operators  transform covariantly\nunder the so--called collinear conformal transformation\n$SL(2,\\mathbb R)$ and are members of infinite dimensional\nconformal multiplets (towers) that are labelled by the conformal\nspin. With other words one takes explicitly or implicitly\nadvantages from the underlying conformal symmetry of the classical\nQCD Lagrangian. If the trace anomaly of the energy--momentum\ntensor, proportional to $\\beta(\\alpha_s)$, is absent, conformal\nsymmetry is present in perturbative QCD and its predictive power\ncan be employed at any order.\n\nIn the following we evaluate the scattering amplitude for the DVCS\nprocess, which is expressed in terms of the time ordered product\nof two electromagnetic currents\n\\begin{eqnarray}\n\\label{Def-HadTen} T_{\\mu\\nu} (q, P_1, P_2) = \\frac{i}{e^2} \\int\ndx e^{i x\\cdot q} \\langle P_2, S_2 | T j_\\mu (x\/2) j_\\nu (-x\/2) |\nP_1, S_1 \\rangle,\n\\end{eqnarray}\nwhere $q = (q_1 + q_2)\/2$ ($\\mu$  and $q_2$ refers to the outgoing\nreal photon). The incoming photon has a large virtuality\n$q_1^2=-{\\cal Q}^2$ and requiring that in the limit $-q^2 = Q^2\\to\n\\infty$ the scaling variables\n\\begin{eqnarray}\n\\xi = \\frac{Q^2}{P\\cdot q}\\,,\\qquad \\eta = -\\frac{\\Delta\\cdot\nq}{P\\cdot q}\\,,\\qquad P = P_1 + P_2\\,, \\qquad \\Delta=P_2-P_1\\,.\n\\end{eqnarray}\nand the momentum transfer squared $\\Delta^2$  are fixed, the\ndominant contributions arise from the light--cone singularities of\nthe time ordered product. In this generalized Bjorken limit one\ncan now employ the OPE to evaluate the hadronic tensor in terms of\nthe leading twist--two operators, where for DVCS kinematics\n$\\eta\\simeq \\xi$ and $Q^2 \\simeq {\\cal Q}^2\/2$.\n\nBefore we outline this step let us introduce a parameterization of\nthe hadronic tensor\n\\begin{eqnarray}\n\\label{decom-T} T_{\\mu\\nu} (q,P,\\Delta) &=& - \\tilde{g}_{\\mu\\nu}\n\\frac{q_\\sigma V^\\sigma}{P\\cdot q} - i \\tilde{\\epsilon}_{\\mu \\nu \\rho\n\\sigma} \\frac{q^\\rho A^\\sigma}{P\\cdot q} + \\cdots.\n\\end{eqnarray}\nTo ensure current conservation, the metric and Levi--Civita tensors\nare contracted here with projection operators, for explicit\ndefinitions of $\\tilde g_{\\mu\\nu}$ and $\\tilde\n\\epsilon_{\\mu\\nu\\rho\\sigma}$ see, e.g., Ref.\\\n\\cite{BelMueNieSch00}. The ellipsis indicates terms that are\nfinally power suppressed in the DVCS amplitude or a determined by\nthe gluon transversity GPD, which is not considered here. We note\nthat in the forward limit $\\Delta\\to 0$ the first and second term\non the r.h.s.\\ are expressed by  the deep inelastic structure\nfunctions $F_1$ and $g_1$. In the parity even sector the vector\n\\begin{eqnarray}\n\\label{dec-FF-V} V^{\\sigma} = \\bar U (P_2, S_2) \\left( {\\cal H}\n\\gamma^\\sigma + {\\cal E} \\frac{i\\sigma^{\\sigma\\rho}\n\\Delta_\\rho}{2M} \\right) U (P_1, S_1) + \\cdots\\, ,\n\\end{eqnarray}\nis decomposed in Compton form factors (CFFs) ${\\cal H}$ and ${\\cal\nE}$, similar for the axial--vector $A^{\\sigma}$ in terms of\n$\\widetilde{\\cal H}$ and $\\widetilde{\\cal E}$, where again higher\ntwist contributions are neglected.\n\nNow we are in the position to employ the OPE.  Let us first\nsuppose that the trace anomaly is absent and so conformal symmetry\ncan be employed. Formally, this can be achieved by assuming that a\nhypothetical fixed point exist, i.e., $\\beta(\\alpha_s^\\ast)=0$. We\ncan then use the COPE, which tells us how the total derivatives\nare arranged. Moreover, it can be considered as a partial wave\nexpansion with respect to the conformal spin $j+2$. For the\ntime--ordered product of two electromagnetic currents it reads in\nthe flavor non--singlet sector as \\cite{FerGriGat71,Mue97a}\n\\begin{eqnarray}\n\\label{Def-COPE}\n \\left\\{T j_\\mu (x) j_\\nu (0)\\right\\}^{\\rm NS} &\\!\\!\\! =\\!\\!\\! &\n\\frac{1}{\\pi^2} \\frac{-\\tilde g_{\\mu \\nu}}{(-x^2 + i 0)^2}\n\\sum_{j=1}^\\infty \\left[1- (-1)^j\\right]\n\\frac{\\Gamma(2-\\gamma_j\/2)(2+j+\\gamma_j\/2)}{\n2^{\\gamma_j\/2}\\Gamma(j+1+\\gamma_j\/2)}\n C_j^{\\rm NS}([-x^2 + i 0] \\mu^2)\n\\nonumber\\\\\n&&\\hspace{3cm}\\times (-2 i x_-)^{j+1} \\int_{0}^{1}\\!du\\,\n[u(1-u)]^{j+1+\\gamma_j\/2} {\\cal O}^{\\rm NS}_j(u x_-) + \\cdots\\,,\n\\end{eqnarray}\nwhere $x_- = \\widetilde n \\cdot x $ with $\\widetilde n^2=0$ is the\nprojection of the vector $x$ on the light cone.  The anomalous\ndimensions of the multiplicatively renormalizable operators ${\\cal\nO}^{\\rm NS}_j$ are denoted as $\\gamma_j(\\alpha_s^\\ast)$. The advantage\nof the COPE is that the Wilson--coefficients (electrical charge\nfactors will be omitted)\n\\begin{eqnarray}\n\\label{Def-WilCoeC} C_j^{\\rm NS}(-x^2 \\mu^2) =\n\\left(-\\mu^2 x^2\\right)^{\\gamma_j\/2}\n\\frac{2^{j+1+\\gamma_j\/2}\\Gamma(5\/2+j+\\gamma_j\/2)}{\n\\Gamma(3\/2)\\Gamma(3+j+\\gamma_j\/2)}\\,\nc_j(\\alpha_s^\\ast)\\,,\n\\end{eqnarray}\nare up to the normalization $c_j(\\alpha_s^\\ast)$ known. The conformal\noperators\n\\begin{eqnarray}\n\\label{Def-ConOpe} {\\cal O}^{\\rm NS}_j(u x_-) =\n\\frac{\\Gamma(3\/2)\\Gamma(1+j)}{2^{j} \\Gamma(3\/2+j)}\\,  (i\n\\partial_+)^j \\bar{\\psi}(u x_-) { \\gamma_+}\\, C_j^{3\/2}\\!\\left(\\!\n\\frac{\\!\\stackrel{\\leftrightarrow}{D}_{+}}{\\partial_+}\\!\n\\right)\\psi(u x_-)\\,,\n\\end{eqnarray}\nare defined with a non--standard normalization. Here\n$\\stackrel{\\leftrightarrow}{D}_+ = \\stackrel{\\rightarrow}{D}_+\n-\\stackrel{\\leftarrow}{D}_+$ and $\\partial_+ =\n\\stackrel{\\rightarrow}{D}_+ + \\stackrel{\\leftarrow}{D}_+$ are the\ncovariant and total derivatives, contracted with the light-like\nvector $n$, i.e., $n^2=0$ and $n\\cdot\\widetilde n =1$, $C_j^{3\/2}$\nare the Gegenbauer polynomials of order $j$ with index $3\/2$. The\nnormalization is chosen so that in the forward limit $\\Delta \\to\n0$ the reduced matrix elements of the conformal operators\n\\begin{eqnarray}\n\\label{Def-ConMomVec}\n\\frac{1}{P_+^{j+1}} \\langle P_2, S_2 \\big|  {\\cal O}_j(0) \\big|P_1,\nS_1 \\rangle= \\frac{1}{P_+}\\bar U (P_2, S_2)\\! \\left(\\! H_j(\\eta,\\Delta^2,\\mu^2)\n\\gamma_+ + E_j(\\eta,\\Delta^2,\\mu^2) \\frac{i\\sigma_{+\\nu}\n\\Delta^\\nu}{2M}\\! \\right) \\! U (P_1, S_1)\n\\end{eqnarray}\ncoincide with the Mellin moments of parton densities. Especially,\nfor odd values of $j$ the forward limit of $H_j$ is the sum of the\nquark and anti--quark  density moments, i.e., $\\lim_{\\Delta\\to\n0}H_j=q_j+\\overline{q}_j$. Moreover, in this limit the\n$u$--integration in Eq.\\ (\\ref{Def-COPE}) can be performed and\nleads to the well--known OPE that is used for the deep inelastic\nscattering structure function $F_1$.  Hence, $c_j(\\alpha_s^\\ast)$\nare identified as the Wilson--coefficients of $F_1$, known to NNLO\norder.\n\n\nEmploying the COPE (\\ref{Def-COPE}), we can now achieve the\nfactorization of the CFFs (\\ref{dec-FF-V}) in a straightforward\nmanner. Plugging Eq.\\ (\\ref{dec-FF-V}) into the definition of the\nhadronic tensor (\\ref{Def-HadTen}), performing Fourier transform,\nand form factor decomposition (\\ref{dec-FF-V}), we arrive for the\nDVCS kinematics $\\eta=\\xi$ at \\cite{Mue97a}\n\\begin{eqnarray}\n\\label{Def-ConParDecInt} \\left\\{{{\\cal H}^{\\rm NS} \\atop {\\cal\nE}^{\\rm NS}} \\right\\}\n= \\sum_{j=1}^\\infty \\frac{1-(-1)^j}{2} {\\xi}^{-j-1} C^{\\rm\nNS}_j(\\mu^2\/Q^2) \\left\\{{H_j^{\\rm NS} \\atop E_j^{\\rm NS}} \\right\\}\n(\\eta, \\Delta^2,\\mu^2)\\Big|_{\\eta=\\xi}\\,.\n\\end{eqnarray}\nHere $H_j^{\\rm NS}$ and $E_j^{\\rm NS}$, cf.\\ Eq.\\ (\\ref{Def-ConMomVec}),\nare given by the conformal moments of the corresponding GPDs\n\\begin{eqnarray}\n\\label{Def-ConGPDmom}\n\\left\\{{H_j^{\\rm NS} \\atop E_j^{\\rm NS}} \\right\\} (\\eta,\n\\Delta^2,\\mu^2)= \\frac{\\Gamma(3\/2)\\Gamma(1+j)}{2^{j}\n\\Gamma(3\/2+j)} \\int_{-1}^{1 }\\! dx\\,\\eta^j\nC_j^{3\/2}\\!\\left(\\frac{x}{\\eta}\\right) \\left\\{{H^{\\rm NS} \\atop\nE^{\\rm NS}} \\right\\} (x,\\eta, \\Delta^2,\\mu^2)\\,.\n\\end{eqnarray}\nNote that these expectation values are\nmeasurable on the lattice \\cite{Hagetal03}.\n\n\nUnfortunately, the series (\\ref{Def-ConParDecInt}) is divergent\nfor $|\\xi| < 1$ and must be resummed, e.g., by means of the\nSommerfeld--Watson transformation \\cite{MueSch05}. Alternatively,\nusing the trick that $\\eta$ is not equated to $\\xi$ allows to deal\nwith Eq.\\ (\\ref{Def-ConParDecInt}) in the analogous manner  as it\nis known from deep inelastic scattering. Namely, a dispersion\nrelation allows to express the Mellin moments of the imaginary\npart by the partial waves, appearing in the COPE\n(\\ref{Def-ConParDecInt}), and thus the inverse  Mellin transform\nprovides the imaginary part of the CFFs. The real\npart can be restored from the dispersion relation, too, and\nfinally equating $\\eta=\\xi$ leads to the same representation as\npresented in Ref.\\ \\cite{MueSch05}:\n\\begin{eqnarray}\n\\label{Def-MelBar} \\left\\{{{\\cal H}^{\\rm NS} \\atop {\\cal E}^{\\rm\nNS}} \\right\\}(\\xi,\\Delta^2,\\mu^2)\n=  \\frac{1}{2 i}\\int_{c-i \\infty}^{c+ i \\infty}\\! dj\\, \\xi^{-j-1}\n\\left[i +\\tan\\left(\\frac{\\pi j}{2}\\right) \\right] C_j^{\\rm\nNS}(Q\/\\mu) \\left\\{{H_j^{\\rm NS} \\atop E_j^{\\rm NS}} \\right\\} (\\xi,\n\\Delta^2,\\mu^2)\\,,\n\\end{eqnarray}\nwhere all singularities of the conformal GPD moments and conformal\nmoments lie on the l.h.s.\\ of the integration path $(c<1)$. Note,\nhowever, that the analytic continuation of  the\nWilson--coefficients (\\ref{Def-WilCoeC}) with respect to the\nconformal spin, analogously done as in deep inelastic scattering,\nleads essentially to an exponential  $2^j$ growing with increasing\n$j$. For $\\xi> 1$, this must be weighed down by a suppression\nfactor that comes from the conformal moments\n(\\ref{Def-ConGPDmom}). This is a rather nontrivial requirement on\ntheir analytic continuation. It should be done in such a way that\nthe integral (\\ref{Def-MelBar}) remains unchanged if the\nintegration contour is closed by an infinite arc, surrounding  the\nfirst and forth quadrant.  The residue theorem states then that\nthis Mellin--Barnes integral is for $\\xi> 1$ equivalent to the\nseries (\\ref{Def-ConParDecInt}). We note that the form of the\nintegrand (\\ref{Def-MelBar}) does in fact not rely on conformal\nsymmetry and so it can be used in any scheme within the\ncorresponding Wilson--coefficients.\n\n\n\nIn the $\\overline{\\rm MS}$ scheme the conformal symmetry is not\nmanifestly implemented in the OPE, however, this failure can be\ncured by a finite renormalization \\cite{Mue97a,BelMue97a}. As\nexplained above, in such a CS scheme and for $\\beta=0$ we can\nsimply borrow the Wilson--coefficients $c_j(\\alpha_s)$ and\nanomalous dimensions $\\gamma_j(\\alpha_s)$ from deep inelastic\nscattering. The inclusion of $\\beta$ proportional terms in Eq.\\\n(\\ref{Def-MelBar}) is, as factorization by itself, conventional.\nTwo possibilities have been discussed in Ref.\\ \\cite{MelMuePas02}.\nNamely, in the CS scheme, one might add the $\\beta$ proportional\nterm from the $\\overline{\\rm MS}$ scheme, evaluated to NNLO in\nRef.\\ \\cite{BelSch98}, while in the $\\overline{\\rm CS}$ scheme the\nrunning of the coupling is implemented in the form of the COPE\n(\\ref{Def-COPE}) in such a way that the Wilson--coefficients\nautonomously evolve in the considered order \\cite{MelMuePas02}. In\nthis letter we prefer the latter convention and, moreover, will\nequate the factorization and renormalization scales, i.e,\n$\\mu=\\mu_f=\\mu_r$. After expansion with respect to $\\alpha_s$, we\nwrite, discarding in the following the superscript NS,  the\nWilson--coefficients (\\ref{Def-WilCoeC}) as\n\\begin{eqnarray}\n\\label{Res-WilCoe-Exp-CS}\nC_\n&\\!\\!\\! =\\!\\!\\!  & \\frac{2^{j+1}\n\\Gamma(j+5\/2)}{\\Gamma(3\/2)\\Gamma(j+3)} \\left[{\\cal C}_j^{(0)} +\n\\frac{\\alpha_s(\\mu)}{2\\pi} {\\cal C}_j^{\n(1)}({\\cal Q}\/\\mu)\n+ \\frac{\\alpha^2_s(\\mu)}{(2\\pi)^2} {\\cal C}_j^{\n(2)}({\\cal Q}\/\\mu) + {\\cal O}(\\alpha_s^3) \\right]\\,,\n\\\\\n\\label{Res-WilCoe-CS-NLO}\n{\\cal C}_j^{\n(0)} &\\!\\!\\! =\\!\\!\\! & 1\\,,\\qquad\n {\\cal C}_j^{\n(1)}\n= c_j^{\n(1) }+\n\\frac{\\gamma_j^{\n(0)}}{2} \\left[s^{(1)}_j+\\ln\\frac{\\mu^2}{{\\cal Q}^2}\\right]\\,,\n\\\\\n\\label{Res-WilCoe-CS-NNLO}\n{\\cal C}_j^\n(2)} &\\!\\!\\! =\\!\\!\\!  & c_j^{\n(2)}+\n\\frac{\\gamma_j^\n(1)}}{2} \\left[ s^{(1)}_j +\n\\ln\\frac{\\mu^2}{{\\cal Q}^2}\\right] + \\frac{\\gamma_j^{\n(0)}}{2} \\left[ c_j^{\n(1)} \\left\\{s^{(1)}_j +\n\\ln\\frac{\\mu^2}{{\\cal Q}^2}\\right\\} + \\frac{\\gamma_j^{\n(0)}}{4} \\right.\n\\\\\n&&\n \\times\\left.\\left\\{s^{(2)}_j + 2 s^{(1)}_j \\ln\\frac{\\mu^2}{{\\cal\nQ}^2}+ \\ln^2\\frac{\\mu^2}{{\\cal Q}^2} \\right\\}\\right]\n-\\frac{\\beta_0}{2}\n \\left[ c_j^{\n(1)}+\n\\frac{\\gamma_j^{\n(0)}}{2} s_j^{(1)} + \\frac{\\gamma_j^{\n(0)}}{4} \\ln\\frac{\\mu^2}{{\\cal Q}^2} \\right]\\ln\\frac{\\mu^2}{{\\cal\nQ}^2}\\,,\n \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\gamma_j^\n(0)}&\\!\\!\\!=\\!\\!\\!& C_F \\left(4S_1(j+1)- 3\n-\\frac{2}{(j+1)(j+2)}  \\right)\\,, \\quad C_F= \\frac{4}{3}\\,,\\\n\\\\\nc_j^\n(1)}&\\!\\!\\!=\\!\\!\\!& C_F \\left[S^2_{1}(1 + j) +\n\\frac{3}{2} S_{1}(j + 2) - \\frac{9}{2}  + \\frac{5-2S_{1}(j)}{2(j +\n1)(j + 2)} -   S_{2}(j + 1)\\right]\\,,\n\\\\\ns_j^{(1)}&\\!\\!\\!=\\!\\!\\!& S_1(j+3\/2)-S_1(j+2) + 2 \\ln(2)\\,, \\quad\ns_j^{(2)}= \\left(s_j^{(1)}\\right)^2 -S_2(j+3\/2)+ S_2(j+2) \\,.\n\\end{eqnarray}\nThe analytic continuation of the harmonic sums are defined by\n$S_1(z)= d\\ln \\Gamma(z+1)\/dz +\\gamma_E$ and $S_2(z)= -d^2\\ln\n\\Gamma(z+1)\/dz^2 + \\zeta(2)$, where $\\gamma_E$ is the Euler\nconstant. The first expansion coefficient of $\\beta(g)\/g =\n(\\alpha_s\/4\\pi) \\beta_0 + {\\cal O}(\\alpha_s^2)$ is $\\beta_0 =\n(2\/3) n_f-11$, where $n_f$ is the number of active quarks. The\ntwo--loop quantities $c_j^{(2)}$ and $\\gamma_j^{(1)}$ are lengthy\nand can be obtained from Ref.\\ \\cite{CurFurPet80,ZijNee92}. The\nevolution of the flavor non--singlet (integer) conformal moments in\nthis $\\overline{\\rm CS}$ scheme is  governed by\n\\begin{eqnarray}\n\\label{Def-RGE} \\mu\\frac{d}{d\\mu} \\left\\{{ H_{j}^{\\rm NS}\\atop\nE_{j}^{\\rm NS}}\\right\\}(\\eta, \\Delta^2,\\mu^2) &\\!\\!\\!=\\!\\!\\!& -\\Bigg[\n\\frac{\\alpha_s(\\mu)}{2\\pi} \\gamma_j^{(0)} +\n\\frac{\\alpha_s^2(\\mu)}{(2\\pi)^2} \\gamma_j^{(1)}+\n\\frac{\\alpha_s^3(\\mu)}{(2\\pi)^3} \\gamma_j^{(2)} +{\\cal\nO}(\\alpha_s^4) \\Bigg] \\left\\{{ H_{j}^{\\rm NS}\\atop E_{j}^{\\rm NS}}\\right\\}(\\eta,\n\\Delta^2,\\mu^2)\n\\nonumber\\\\\n&&\\hspace{0.5cm} -\\frac{\\beta_0}{2}\n\\frac{\\alpha_s^3(\\mu)}{(2\\pi)^3}\\sum_{k=0}^{j-2} \\eta^{j-k}\n\\left[\\Delta_{jk}^{\\overline{{\\rm CS}}}+{\\cal O}(\\alpha_s)\n\\right]\\left\\{{ H_{k}^{\\rm NS}\\atop E_{k}^{\\rm NS}}\\right\\}(\\eta, \\Delta^2,\\mu^2)\\,,\n\\end{eqnarray}\nwhere the mixing matrix $\\Delta_{jk}^{\\overline{{\\rm CS}}}$ is not\ncompletely known.\n\n\n\nWe remark that the NLO corrections in the $\\overline{\\rm MS}$\nscheme can be easily evaluated from the conformal moments of the\nhard--scattering amplitude, e.g., given in Ref.\\\n\\cite{BelMueNieSch99}. All integrals, which are needed, are given\nin Appendix C of Ref.\\ \\cite{MelMuePas02} for integer conformal\nspin. The analytic continuation is straightforward and so in the\n$\\overline{\\rm MS}$ scheme Eq.\\ (\\ref{Res-WilCoe-CS-NLO}) is to\nreplace by\n\\begin{eqnarray}\n\\label{Res-WilCoe-MS-NLO} {\\cal C}_j^{ {\\overline{\\rm MS}}\n(1)}&\\!\\!\\!=\\!\\!\\!& C_F \\left[2 S^2_{1}(1 + j)- \\frac{9}{2}  +\n\\frac{5-4S_{1}(j+1)}{2(j + 1)(j + 2)} +\n\\frac{1}{(j+1)^2(j+2)^2}\\right] + \\frac{\\gamma_j^{\n(0)}}{2}\n\\ln\\frac{\\mu^2}{{\\cal Q}^2} \\,.\n\\end{eqnarray}\nIn this scheme also the complete anomalous dimension matrix is\nknown to two--loop accuracy \\cite{Mue94}.\n\n\n\n\n\\section{Numerical results}\n\\label{Sec-NumRes}\n\nIn this Section we numerically study the   radiative corrections\nto the CFF\n\\begin{eqnarray}\n\\label{Def-ComForFacH}\n{\\cal H}^{{\\rm N}^P{\\rm LO}\n=\n\n\n \\!\\sum_{p=0}^P\n \\left(\\!\\frac{\\alpha_s(\\mu)}{2\\pi}\\!\\right)^p\n\\frac{1}{2 i}\\!\\int_{c-i\\infty }^{c+i\\infty }\\!\\!\\!dj\\, \\xi^{-j-1}\n\\frac{2^{j+1}\\Gamma(5\/2+j)}{\\Gamma(3\/2)\\Gamma(3+j)} \\left[\\!\ni+\\tan\\!\\left(\\!\\frac{\\pi j}{2}\\!\\right)\\!\\right]\n {\\cal C}^{(p)}_j\\!\\!\\left(\\!\\frac{{\\cal\nQ}}{\\mu}\\!\\right)\nH_j^{\\rm NS}(\\xi,\\Delta^2,\\mu^2 )\n\\end{eqnarray}\nat NLO $(P=1)$ and NNLO $(P=2)$ accuracy. Certainly, the size of\nthe radiative corrections depend on the GPD distribution itself.\nAs a GPD model assumption, we suppose that the expansion of\n$H_j^{\\rm NS}$ in powers of $\\xi^2$  induces a systematic\nexpansion of the CFF (\\ref{Def-ComForFacH}). A closer look to this\nissue has been given in Sect.\\ 4 of Ref.\\ \\cite{MueSch05}. The GPD\nmoments are generically parameterized as\n\\begin{eqnarray}\n\\label{Def-BuiBlo} H^{\\rm NS}_j(\\Delta^2,\\xi,\\mu_0^2)= F^{\\rm NS}(\\Delta^2)\n\\frac{B(1-\\alpha(\\Delta^2)+j,\\beta+1)}{B(1-\\alpha(\\Delta^2),\\beta+1)}\n+ {\\cal O}(\\xi^2)\\,.\n\\end{eqnarray}\nIn the forward limit, i.e., $\\Delta\\to 0$,  these moments reduce\nto the Mellin moments of the unpolarized parton density. Hence,\nthe parameters $\\alpha$ and $\\beta$ characterize the small and\nlarge $x$ behavior, respectively, i.e., $q(x,\\mu_0)\\propto\nx^{-\\alpha} (1-x)^\\beta$. For non-singlet parton densities the\ngeneric values are $\\alpha(\\Delta^2=0)=1\/2$ and $\\beta=3$.  In the\noff--forward kinematics we consider $\\alpha(\\Delta^2) = \\alpha(0) +\n\\alpha^\\prime(0)\\Delta^2 $ as a linear meson Regge trajectory with\n$\\alpha(0) = 1\/2$ and $\\alpha^\\prime(0) = 0.9\\, \\mbox{\\rm\nGeV}^{-2}$. Note also that the conformal GPD moment with $j=0$\nreduces to the elastic form factor $F^{\\rm NS}(\\Delta^2)$.\nCertainly, a more realistic ansatz of GPD moments is formed by a\nlinear combination of the building blocks (\\ref{Def-BuiBlo}).\n\n\\begin{figure}[t]\n\\begin{center}\n\\mbox{\n\\begin{picture}(450,240)(0,0)\n\\put(175,240){(a)} \\put(10,130){\\insertfig{7.1}{FigNNLO1a}}\n\\put(-10,145){\\rotatebox{90}{$K_{\\rm abs}({\\cal Q}^2= 2.5\\,\n{\\rm GeV}^2)$}}\n \\put(430,240){(b)}\n\\put(260,130){\\insertfig{7.1}{FigNNLO1b}}\n \\put(175,110){(c)}\n\\put(10,0){\\insertfig{7.1}{FigNNLO1c}} \\put(205,-5){$\\xi$}\n\\put(430,110){(d)} \\put(260,0){\\insertfig{7.1}{FigNNLO1d}}\n\\put(-10,15){\\rotatebox{90}{$K_{\\rm arg}({\\cal Q}^2= 2.5\\,\n{\\rm GeV}^2)$}} \\put(455,-5){$\\xi$} \\put(66,107){\\tiny NNLO,\\,\n$\\beta_0=-9$} \\put(66,94){\\tiny NNLO,\\, $\\beta_0=0$}\n\\put(66,82){\\tiny NLO, $\\overline{\\rm CS}$} \\put(66,69){\\tiny NLO, $\\overline{\\rm\nMS}$}\n\\end{picture}\n}\n\\end{center}\n\\caption{ \\label{FigNNLO} The relative radiative corrections,\ndefined in Eq.\\ (\\ref{Def-Rrat}), are plotted versus $\\xi$ for the\nmodulus [(a) and (b)] and phase [(c) and (d)] of the  CFF\n(\\ref{Def-ComForFacH}) with $\\alpha=0.5$ [(a) and (c)] and\n$\\alpha= -0.1$ [(b) and (d)]: NNLO in full (solid) and $\\beta_0=0$\n(dash--dotted) as well as in NLO for the $\\overline{\\rm CS}$ (dashed) and\n$\\overline{\\rm MS}$ (dotted) scheme. We  always set $\\mu=\n{\\cal Q}$ and  $\\alpha_s({\\cal Q}^2= 2.5\\, {\\rm GeV}^2)\/\\pi = 0.1$. }\n\\end{figure}\nLet us first compare the radiative corrections in the\n$\\overline{\\rm MS}$ and $\\overline{\\rm CS}$ scheme to NLO accuracy. Strictly\nspoken, at a given input scale ${\\cal Q}_0$ there is no difference\nbetween both predictions, if the non--perturbative quantities are\ntransformed, too,\nand a consequent expansion in $\\alpha_s$ is performed.  Usually,\nthe GPD moments are taken from some non--perturbative (model)\ncalculation or ansatz, where the matching with the perturbative\nprescriptions has its own uncertainties. So let us take the same\ninput in both schemes and study the relative changes\n\\begin{eqnarray}\n\\label{Def-Rrat} K^P_{\\rm abs}=\\frac{\\left|{\\cal H}^{{\\rm N}^P{\\rm\nLO}}\\right|}{\\left|{\\cal H}^{{\\rm N}^{P-1}{\\rm LO}}\\right|}\\,,\n\\qquad {K}^P_{\\rm arg}= \\frac{{\\rm arg}\\!\\left(\\! {\\cal H}^{{\\rm\nN}^P{\\rm LO}}\\!\\right)}{{\\rm arg}\\!\\left( {\\cal H}^{{\\rm\nN}^{P-1}{\\rm LO}}\\right)}\n\\end{eqnarray}\nto the modulus and phase of the CFF (\\ref{Def-ComForFacH}). These\nquantities do not suffer from large radiative corrections as it is\nartificially the case for the real part of the\namplitude\\footnote{\\label{Foo-DefRadCor}The real part possesses a\nzero in the valence quark region, which position depends on  the\napproximation. Hence, in the vicinity of this zero the relative\nradiative corrections blow up.}. One should bear in mind that\nthese factors are a measure for the necessary reparameterization\nof the GPD when one includes the next order in a given scheme. In\nFig.\\ \\ref{FigNNLO} we depict for the typical kinematics in fixed\ntarget experiments, i.e., $0.05\\lesssim \\xi \\lesssim 0.3 $, the\n$K$ factors to NLO as dashed and dotted lines for the\n$\\overline{\\rm CS}$ and $\\overline{\\rm MS}$ schemes, respectively.\nWe set $\\mu={\\cal Q }$ and independent of the considered\napproximation we choose $\\alpha_s(\\mu_r^2= 2.5\\,{\\rm GeV}^2 ) =\n0.1 \\pi$. From the panels (a) and (b) it can been realized that in\nthe $\\overline{\\rm MS}$ scheme the radiative corrections to the\nmodulus are up to 20\\% and 30\\% for $\\alpha=0.5$ ($\\Delta^2=0$)\nand $\\alpha=-0.1$ ($\\Delta^2=-0.\\overline{6}\\; {\\rm GeV}^2 $),\nrespectively. In the $\\overline{\\rm CS}$ scheme these radiative\ncorrections  are reduced by 30\\%. Note that such a reduction has\nbeen observed in a quite different processes, namely, for the\nphoton--to--pion transition form factor\\footnote{This process is\nalso evaluated within the OPE and the  reduction of NLO\ncorrections has a common origin. Namely, in the $\\overline{\\rm\nCS}$, compared to the $\\overline{\\rm MS}$, scheme the first few\nWilson--coefficients are smaller, while the stronger logarithmic\ngrowing with increasing $j$ is anyway suppressed by the\nnon--perturbative input, see Eqs.\\ (\\ref{Res-WilCoe-CS-NLO}) and\n(\\ref{Res-WilCoe-MS-NLO}).}. The relative radiative corrections to\nthe phase is in both cases of about 15\\% at $\\xi=0.6$ and\ndiminishes with decreasing $\\xi$. These findings qualitatively\nagree with previous ones in which the Radyushkin ansatz for GPDs\nwas used \\cite{BelMueNieSch99}.\n\n\nWe study now the radiative corrections to NNLO accuracy. To\nsimplify their evaluation, we take for $c_j^{(2)}$ a fit, given in\nRef.\\ \\cite{NeeVog99}, rather than the exact expression. For the\n$\\beta_0$ proportional term we have checked that within\n$|c|=|\\Re{\\rm e} j| \\le 1\/2$, see Mellin--Barnes integral\n(\\ref{Def-ComForFacH}), the accuracy is on the level of one per\nmill or better. Outside this region, the deviation can be larger\nand one might use Fortran routines \\cite{BluMoc05}.  For three\nquark flavors $n_f=3$, the same scale settings, and the initial\nvalue of $\\alpha_s$ as specified above for NLO,  we plot in Fig.\\\n\\ref{FigNNLO} the ratios (\\ref{Def-Rrat}) in the $\\overline{\\rm\nCS}$ scheme for $P=2$ as solid line. The $\\xi$--dependence of the\nmodulus $K$--factor, see panels (a) and (b), is rather flat and\nthe modulus decreases on the 5\\% level. The radiative correction\nto the phase is again negligible for smaller values of $\\xi$ and\nincreases now only to 5\\% at $\\xi=0.6$. We note that the $\\beta_0$\ninduced corrections are about twice times larger than the\nremaining ones and are opposite in sign, see dash--dotted line.\nRemarkably, for $\\beta_0=0$ we find then  an opposite behavior of\nthe $K$ factors as in NLO. This arise from a sign alternating\nseries and as for the photon--to--pion transition form factor, this\nmight be  considered as a reminiscence on Sudakov double logs\n\\cite{MusRad97}.\n\n\nLet us finally address the modification of the scale dependence\ndue to the higher order corrections. Note that the analysis about scheme\ndependence in Ref.\\ \\cite{Mue98} suggests that the discrepancy\nbetween the $\\overline{{\\rm CS}}$ and $\\overline{{\\rm MS}}$\nschemes at NLO accuracy are mainly induced by the\nWilson--coefficients while the evolution yields minor differences.\nSo we only consider here the $\\overline{{\\rm CS}}$ scheme and\nanalogously  as in  Eq.\\ (\\ref{Def-Rrat}), we quantify the\nrelative changes of $d{\\cal H}\/d\\ln{\\cal Q}^2$ by the ratios\n\\begin{eqnarray}\\label{Def-Rrat-dot}\n\\dot{K}^P_{\\rm abs}= \\left|\\frac{d {\\cal H}^{\\rm {N}^P{\\rm\nLO}}}{d\\ln{\\cal Q}^2}\\right|\\Bigg\/\\left|\\frac{d {\\cal H}^{\\rm\n{N}^{P-1}{\\rm LO}}}{d\\ln{\\cal Q}^2}\\right|\\,, \\; \\dot{K}^P_{\\rm\narg}=\n  \\left[\\pi +{\\rm arg}\\!\\left(\\!\n  \\frac{d\\, {\\cal H}^{\\rm {N}^P{\\rm LO}}}{d\\ln{\\cal Q}^2}\\!\\right)\n  \\right]\\Bigg\/\n  \\left[\\pi +{\\rm arg}\\!\n  \\left(\\! \\frac{d\\, {\\cal H}^{\\rm {N}^{P-1}{\\rm LO}}}{d\\ln{\\cal\n  Q}^2}\\!\\right)\\right].\n\\end{eqnarray}\nBoth the numerator and denominator in the latter $\\dot{K}$--factor\nare now defined in the interval $[0,2\\pi]$ and so the appearance\nof a zero in the denominator is avoided, cf.\\ footnote\n\\ref{Foo-DefRadCor}. We take the same scale setting and initial\ncondition for the (exact) evolution of $\\alpha_s({\\cal Q})$ as\nabove. The conformal moments (\\ref{Def-BuiBlo}) are evolved in the\n$\\overline{\\rm CS}$ scheme.  The input scale is now $\\mu_0^2= 0.5\\,\n{\\rm GeV}^2$, which is typical for the matching of perturbative\nand non--perturbative QCD. The non--leading logs in the solution of\nthe evolution equation (\\ref{Def-RGE}) are expanded with respect\nto $\\alpha_s$ and are consistently combined with the\nWilson--coefficients (\\ref{Res-WilCoe-Exp-CS}) in the considered\norder. Here the forward anomalous dimensions are taken from Ref.\\\n\\cite{MocVerVog04} and the unknown NNLO mixing term\n$\\Delta_{jk}^{\\overline{{\\rm CS}}}$ in Eq.\\ (\\ref{Def-RGE}) is\nneglected. This mixing can be suppressed at the input scale within\nan appropriate initial condition  and so we expect only a minor\nnumerical effect; see also Ref.\\\nRef.\\ \\cite{Mue98}.\n\\begin{figure}[t]\n\\begin{center}\n\\mbox{\n\\begin{picture}(450,120)(0,0)\n \\put(175,100){(a)}\n\\put(10,0){\\insertfig{7.2}{FigNNLO2a}} \\put(205,-5){$\\xi$}\n\\put(425,100){(b)} \\put(260,0){\\insertfig{7.1}{FigNNLO2b}}\n\\put(-15,20){\\rotatebox{90}{$\\dot{K}^{\\rm abs}({\\cal Q}^2= 4\\,\n{\\rm GeV}^2 ) $}} \\put(235,20){\\rotatebox{90}{$\\dot{K}^{\\rm arg}{\\cal\nQ}^2=4\\,{\\rm GeV}^2 ) $}} \\put(455,-5){$\\xi$} \\put(315,116){\\tiny\nNNLO,\\, $\\alpha=0.5$} \\put(315,104){\\tiny NLO,\\,\\phantom{N}\n$\\alpha=0.5$} \\put(315,92){\\tiny NNLO,\\,\n $\\alpha=-0.1$} \\put(315,81){\\tiny NLO,\\,\\phantom{N}\n$\\alpha=-0.1$}\n\\end{picture}\n}\n\\end{center}\n\\caption{ \\label{Fig-ScaDep} The relative change of scale\ndependence, cf.\\ Eq.\\ (\\ref{Def-Rrat-dot}), in the $\\overline{\\rm\nCS}$ scheme at NLO (dashed, dotted)  and NNLO (solid, dash--dotted)\nversus $\\xi$ is depicted for the modulus (a) and phase (b) of the\nCFF (\\ref{Def-ComForFacH}) with $\\alpha=0.5$ (dashed, solid)  and\n$\\alpha= -0.1$ (dotted, dash--dotted) and ${\\cal Q}^2 =4\\, {\\rm GeV}^2$.\nWe set $\\mu = {\\cal Q}$, $\\alpha_s(\\mu_r^2= 2.5\\, {\\rm GeV}^2 ) \/\\pi =\n0.1$ and took the input (\\ref{Def-BuiBlo}) at the scale ${\\cal\nQ}^2_{0} = 0.5\\, {\\rm GeV}^2 $. }\n\\end{figure}\nThe dashed and dotted lines in Fig.\\ \\ref{Fig-ScaDep} show that in\nNLO the scale dependence changes of about 30\\% to 50\\% for the\nmodulus and up to 20\\% for the phase. The latter $\\dot{K}$--factor\nbecomes close to one for smaller values of $\\xi$. The NNLO\ncorrections, compared to the full NLO result, are milder. We\nobserve (solid and dash--dotted line) a 20\\% or less and a maximal\n10\\% correction for the modulus and phase, respectively.\n\n\n\n\\section{Conclusions}\n\\label{Sec-Con}\n\nIn this letter we have taken the first steps in the full investigation\nof radiative NNLO corrections to the DVCS scattering\namplitude. Thereby, we employed the COPE, which allows an\neconomical treatment of perturbative corrections, and restricted\nourselves to the CFF $\\cal H$ in the flavor non--singlet sector.\nThis study can be immediately adopted for $\\cal E$. The extension\nto the axial vector case, i.e., to $\\widetilde {\\cal H}$ and\n$\\widetilde {\\cal E}$, is straightforward, however, here the anomalous\ndimensions to three loop order cannot be borrowed from the\npolarized DIS results. The conformal  approach can be also\nstraightforwardly extended to the flavor singlet sector, which\nwill be presented somewhere else.\n\nWe relied on GPDs for which the expansion of the conformal moments\nin powers of $\\xi^2$ is meaningful and used in our analysis only\nthe leading term. Whether this assumption is  too restrictive is\nan open problem, which might be resolved by lattice calculations\n\\cite{Hagetal03}. At least there is a warning from perturbative\nQCD. Namely, in the $\\overline{\\rm MS}$ scheme $\\xi^2$--suppressed\ncontributions in the Wilson--coefficients are resummed and lead to\nlarger radiative corrections than in the $\\overline{\\rm CS}$ one.\nLet us summarize our numerical findings for fixed target\nkinematics, i.e., $0.05 \\lesssim\\xi \\lesssim 0.3$:\n\\begin{itemize}\n\\item Compared to\nthe $\\overline{\\rm MS}$ scheme, in the $\\overline{\\rm CS}$ one the\nradiative NLO corrections to the modulus of the CFF are reduced by\n30\\% or so, while for the phase no significant differences appear.\n\n\\item The relative NNLO corrections compared to the NLO ones are\nof the 5\\% level or below, where the $\\beta_0$ proportional\nones dominate and determine the sign.\n\n\\item The change of the  scale dependence due to the NLO corrections\nis rather large and is for the modulus (phase) of the CFF of about\n30\\% to 40\\% (15\\% or smaller). Comparing NNLO to NLO, the\ntwo--loop corrections are reduced to less than 20\\% (10\\%) .\n\\end{itemize}\nLet us add that the\nWilson--coefficients in the vector and axial--vector channel  possess a\nsimilar $j$ dependence. So we might expect that for\n$\\widetilde {\\cal H}$  and $\\widetilde {\\cal E}$ (axial--vector\ncase) comparable radiative corrections appear.\nWe conclude that our findings support the perturbative description\nof the DVCS process at scales which are accessible in the\nkinematics of present fixed target experiments.\n\n{\\ }\n\n\\noindent\n{\\bf Acknowledgement}\n\n\\noindent\nFor discussions I am indebted to A.\\ Manashov, B.~Meli{\\' c}, K.~Kumeri{\\v c}ki,\nK.~Passek--Kumeri{\\v c}ki,  and A.\\ Sch\\\"afer. This project has been\nsupported by the Deutsche Forschungsgemeinschaft and the\nU.S. National Science Foundation under grant no. PHY--0456520.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nModeling materials in the electronic ground-state requires an accurate description of the Born-Oppenheimer potential energy surface (PES). First-principles models based on density functional theory (DFT) have good predictive power, but high computational complexity. Empirical force fields are more efficient but often less accurate than DFT, and are typically not applicable when chemical bonds change within a system of interest. \nPES models based on machine-learning (ML) offer a promising route to overcome the dilemma of accuracy versus efficiency~\\cite{behler2007generalized,bartok2010gaussian,rupp2012fast,bartok2013representing,thompson2015spectral,shapeev2016moment,schutt2017quantum,smith2017ani,chmiela2017machine,chmiela2018towards,yao2018tensormol,han2018deep,zhang2018deep,zhang2018end,lubbers2018hierarchical,unke2019physnet}. \nThese models combine force field efficiency with first-principles accuracy and have been successful \nin a number of applications that could not be tackled with direct\nfirst principles studies. \nHere, we focus on one such method, the deep potential (DP) scheme~\\cite{zhang2018deep,zhang2018end}, which, upon training with DFT data, can achieve \naccuracy comparable to that of \\textit{ab initio} molecular dynamics (AIMD). On high-performance computer platforms, deep potential molecular dynamics (DPMD) simulations can model millions of atoms on timescales of tens of nanoseconds or even longer~\\cite{lu202186,jia2020pushing}. \nSuccessful applications to date include studies of\nthe signatures of a liquid-liquid phase transition in supercooled water~\\cite{gartner2020signatures}, nucleation of crystalline Si from the melt~\\cite{bonati2018silicon}, \nphase diagrams of Ga~\\cite{niu2020ab} and  water~\\cite{zhang2021phase}, \nstructural motifs in the growth of quasicrystals~\\cite{han2020dynamic}, the reactive uptake of \\chem{N_2O_5} in interfacial processes~\\cite{galib2021reactive},\nreactive processes in combustion~\\cite{zeng2020complex}, \nthe dissociative chemisorption of water at the water-titania interface~\\cite{andrade2020free}, and of the 1D cooperative diffusion in a simple cubic crystal~\\cite{wang2021electronically}.\n\n\nMost ML models, including DP, assume that the PES can be represented by a sum of atomic terms describing the many-body interactions among the atoms within some fixed cutoff distance. This construction is computationally advantageous as its complexity scales linearly with system size. On physical grounds, however, it entails a severe approximation, because electrons and nuclei are charged particles that experience long-range Coulomb forces. In spite of that, local effective interaction models work well in many cases. In metals, this can be attributed to complete electronic screening. In insulators, screening is incomplete, yet, the interatomic interactions are often represented\nsatisfactorily by short-range ML models. These models are trained on DFT data, which include long-range electrostatics, suggesting\nthat some features of the long-range interactions may be mimicked by short-range models. \nWhile this treatment is sufficient in many cases, particularly when dealing with systems that are, on average, homogeneous and neutral at the molecular scale, failures of the short-range models in cluster, interfacial and vapor phase properties are well known~\\cite{yue2021short,niblett2021learning}.\nA quintessential example of a property\nthat cannot be simulated by short-range models is the longitudinal-transverse splitting of the long-wavelength\noptical phonon modes in polar crystals. Long-range electrostatics should be even more important in heterogeneous systems, where \ncharge separation at large scale may be induced by external fields or by chemical potentials. Examples include electrolyte solutions \nin contact with electrodes~\\cite{scalfi2021molecular},\ndipolar surfaces~\\cite{patra2003molecular,wohlert2004range}, protein folding problems~\\cite{zhou2018electrostatic}, and the binding affinity of ligands to proteins~\\cite{rocklin2013calculating}. \n\n\nA possible way of incorporating long-range electrostatics within ML models is to assume \nthat the PES has two contributions. One accounts \nfor short-range interactions and is constructed as in standard ML models. The other accounts for\nlong-range interactions and is approximated by the electrostatic energy of a system of partial point charges located at the atom sites, \nwith the added constraint of global charge conservation. \nThe effective charges can be either defined empirically, as commonly done in the context of \nforce-field models ~\\cite{deng2019electrostatic}, or can be found by machine learning~\\cite{artrith2011high,bereau2015transferable,yao2018tensormol,bereau2018non,unke2019physnet},\nin which case the partial charges are matched to reference charges extracted from DFT calculations by\npartitioning the electronic charge density among the atoms~\\cite{artrith2011high,bereau2015transferable,bereau2018non}. \nA difficulty with this approach is that an atom in a molecule, or a material, does not have a well defined charge, because the charge densities of neighboring atoms overlap.\nAs a consequence, different partitioning schemes may lead to different results~\\cite{nebgen2018transferable,sifain2018discovering}. \nTo reduce this ambiguity, charge partitioning schemes have been combined with the concepts of electronegativity and hardness~\\cite{ghasemi2015interatomic}, \nwhich, however, also lack a rigorous definition. \nInterestingly, the accuracy and transferability of molecular ML models with effective atomic charges \nwere found to improve significantly when the molecular dipole moment was added to energy and forces in the training data~\\cite{ghasemi2015interatomic,ko2021fourth,ko2021general}. \n\\recheck{This condition is implemented in PhysNet~\\cite{unke2019physnet}, a model in which the partial charges are trained to best reproduce total energy and dipole of a system.}\nA more flexible representation than the point charge models \nis provided by the long-distance equivariant (LODE) framework, \nin which the charge density is approximated by a sum\nof atom centered spherical Gaussians, or other localized functions~\\cite{grisafi2019incorporating,grisafi2021multi}.\nLODE successfully describes the mutual interaction of charged molecular species at large separation, a \nproperty beyond the reach of short-range ML models.\n   \nThe above schemes can be seen as different realizations of a coarse-graining transformation that approximates   \nthe electronic density with a sum of atom centered spherical contributions. \n\\recheck{This representation can provide a good overall fit of the charge density and of the total dipole of a molecule, but, in general, does not describe correctly the dipolar fluctuations that couple with the electric field generated by distant charges in extended systems within periodic boundary conditions.\nThe reason is that the dipole of a periodic crystal can only be defined modulo a quantum, and the polarization fluctuations typically include dipolar fluxes across the cell boundary, originating from the delocalized nature of the quantum mechanical electronic charge distribution. \nThese fluxes are related to the phase of the wavefunction and cannot be described, in general, by partial atomic charges (see e.g., Ref.~\\citen{vanderbilt2018berry}). \nInterestingly, an exception to this rule occurs when a crystal is made of nonoverlapping molecular units, because, in this case, the total dipole is the sum of the molecular dipoles.}\n\n\n\n\nIn this work, we propose an alternative deep learning model for insulating systems that overcomes this limitation.\nAs in previous approaches, we assume that the PES has short- and long-range contributions. \nThe short-range contribution has the standard form of the DP model. The long-range contribution is the electrostatic energy of a system of spherical Gaussian charges, associated with the ions (nuclei+frozen core electrons) and the valence electrons, located, respectively, at the ionic and electronic sites. The latter are defined by the averages of the positions of the maximally localized Wannier centers~\\cite{marzari2012maximally} associated with specific atoms~\\cite{zhang2020deep}. \nIn general, ionic and electronic sites do not coincide. The electronic site coordinates are not independent dynamical variables, but are fixed by the atomic configuration, as required by the Born-Oppenheimer adiabatic separation of ionic and electronic dynamics.\n\\recheck{We assume that the electronic sites, hereafter also called Wannier centroids, depend only on the atoms in their neighborhood, a conjecture that can be rationalized in terms of the so-called nearsightedness of electronic matter~\\cite{prodan2005nearsightedness}. \nWe found that this condition is well satisfied in all systems we have studied to date.}\nThus, the environmental dependence of the centroids can be modeled accurately by a deep neural network like deep Wannier (DW), \nwhich was introduced in Ref.~\\citen{zhang2020deep} to describe the dielectric response of insulators.  \nBy combining DP and DW we construct a model of the PES that includes explicit long-range electrostatic interactions. \nThe new scheme, called deep potential long-range (DPLR), has several advantages relative to atom centered models. \nFirst, the centroid distributions have integer charges and automatically conserve the total charge of the system. Second,\ntheir positions derive from a unitary transformation in the subspace of the occupied Kohn-Sham orbitals. \n\\recheck{By construction, this representation rigorously describes molecular dipoles in finite systems and dipolar fluctuations in condensed media (see e.g.~Ref.~\\citen{vanderbilt2018berry}).}\nThe spherical Gaussian distributions adopted in DPLR differ from the maximally localized Wannier distributions in the quadrupole and higher moments. This results in a potential energy error that converges well with size. In contrast,\nthe errors of atom centered distributions start at the dipole level, and the potential energy error is only conditionally convergent with \nsize. \nThe DPLR model is physically \nmeaningful, symmetry preserving, and smooth, so that forces and virial can be analytically calculated, preserving the conservative properties of the adopted molecular dynamics scheme. \nThe approach is also computationally efficient, as fast algorithms \nfor calculating Ewald sums for Gaussian charges under periodic boundary conditions are well established~\\cite{ewald1921die,hockney1988computer,darden1993pme,essmann1995spm}. DPLR describes polarization fluctuations in finite and extended systems. Thus, it also allows us to model the response of a system to an externally applied field. \n\n\nWhile preparing this manuscript, we became aware of a\nrecent preprint by Gao and Remsing~\\cite{gao2021self}, in which the\nelectronic structure information encoded in the Wannier centers is used to construct a ML model of the PES including long-range electrostatics. This model, called self-consistent field neural network (SCFNN), differs from DPLR because the separation between short- and long-range\ncontributions is not done in the way in which the training data are handled, but in the way in which they are generated. \nThe model requires short-range DFT data obtained, in principle, from calculations with a truncated Coulomb potential. \nA drawback of this formulation is that it introduces a self-consistency condition for the Wannier center positions. \nAs a consequence, the simplicity of the ML construction is lost, and the dynamics of the model is no longer conservative, unless\nformulations such as those proposed by Car and Parrinello are introduced~\\cite{car1985unified}. In practice, separating standard DFT data into data\nfor a truncated Coulomb potential and data for its long-range counterpart is not straightforward, and the authors overcome this difficulty by invoking linear response conditions. \n \n\nIn this manuscript, we focus on DPLR and\nhow it compares to the original DP model.\nTheory and method are presented in\nSect.~II. In Sect.~III we construct the DP, DW, and DPLR models for water that we use in the applications discussed in Sect.~IV.\nIn that section we compare DPLR, DP, and direct DFT calculations for the binding energy curve of the water dimer and for the free energy profile of a water molecule as a function of the distance from a liquid water slab.\nIn Sect.~V we construct DP and DPLR models for small atomic displacements around the classical ground-state in crystalline NaCl, and use these models to compute the phonon dispersion curves.\nWe show that the non-analytic contribution to the dynamical matrix, arising from long-range dipole-dipole interactions, can be calculated with DW and its polarizability extension~\\cite{sommers2020raman}. \nWe then study the evolution of the analytical phonon dispersion curves calculated with finite supercells of increasing size. \nWhile the modes calculated with the DP model do not exhibit size dependence, those calculated with DPLR depend on size, and partially recover the longitudinal-transverse splitting of the optical modes, within the limitations of the finite supercells used in the calculations. \\recheck{In Sect. VI we discuss how the model can be used to study the response of a system to an externally applied electric field.}  Finally, Sect.~VII is devoted to our conclusions.  \n\n\n\n\\section{Theory and method}\n\nWe focus on extended systems modelled with a periodically repeated supercell. Finite systems can be investigated in this way, provided the supercell is large enough that interactions between periodic images can be neglected. With periodic boundary conditions the system must be electrically neutral. However, the scheme could be easily extended to different choices of boundary conditions.     \n\n\\subsection{Electrostatic energy}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.4\\textwidth]{fig-wc.pdf}\n    \\caption{The Wanner centroid (WC) associated with a water molecule is represented by the purple dot. The maximally localized Wannier Centers (MLWCs) corresponding to bond pairs and lone pairs are represented by green and blue dots, respectively. The WC is the geometric center of the 4 MLWCs. In molecular dynamics trajectories of water systems the WCs are uniquely associated with the oxygen atoms.}\n    \\label{fig:wc}\n\\end{figure}\n\nIn the reference DFT model we adopt a pseudopotential framework. Thus, in what follows, electrons stand for valence electrons, and ions for nuclei plus frozen core electrons.\nHowever, the method would be applicable to all-electron DFT methods as well, in which case ions would stand for nuclei and electrons would include valence and core. \nWe indicate the charge density of ions and electrons by $\\rho_{\\textrm{ion}}(r)$ and $\\rho_e(r)$, respectively. When the system is in the ionic configuration\n$\\{R_I\\}$, $\\rho_{\\textrm{ion}}(r)$ is given by: \n\\begin{align}\n    \\rho_{\\textrm{ion}}(r) = \\sum_I \\rho_I (r-R_I),\n\\end{align}\nwhere $R_I$ indicates ion coordinates, and $\\rho_I (r) = q_I \\delta (r)$, in terms of $q_I$, the charge of the ion $I$, and of $\\delta (r)$, the Dirac delta distribution. In insulators, the density of the electrons, $\\rho_e(r)$, can be represented by a sum of local distributions \nvia a unitary transformation of the occupied Bloch orbitals onto maximally localized Wannier functions~\\cite{marzari2012maximally}. The centers of the local distributions are called maximally localized Wannier centers (MLWC). \n\\recheck{Often, during molecular evolution}, the same MLWCs can be uniquely\nassigned to the same atom along a trajectory.\nFor example, in water systems the same four MLWCs are always nearest \nto the same oxygen atom irrespective of liquid diffusion and molecular dissociation. It is then convenient to define the $n$-th Wannier centroid (WC), $W_n$, as the average of\nthe MLWCs assigned to a given atom, as illustrated in Fig.~\\ref{fig:wc} for a water molecule. \n\\recheck{Lumping into a single centroid associated to the O atom the four Wannier centers of a water molecule has the following advantages: (i) it makes straightforward to satisfy, within the deep neural network (DNN) model, the Born-Oppenheimer constraint of parametric dependence of the ground-state electronic properties on the atomic coordinates; (ii) it eliminates the need for imposing permutational symmetry between equivalent Wannier centers in a molecule; (iii) it is computationally efficient.\nWe note, however, that a unique association of the Wannier centers to specific atoms is not possible in presence of electron transfer between different atoms. Dealing with these events requires a significant generalization of the method. Ignoring this possibility,}\n$\\rho_e (r)$ takes the form:           \n\\begin{align}\n    \\rho_e(r) = \\sum_n \\rho_n(r-W_n),\n\\end{align}\nwhere $\\rho_n$ is the sum of the maximally localized Wannier distributions associated with the $n$-th WC. \nFor simplicity, we consider\nspin saturated systems, in which each maximally localized Wannier distribution carries a charge of 2 electrons ($-2e$). Thus, $\\rho_n$ carries an integer charge $q_n$, equal to $-2e$ times the number of Wannier functions associated with the $n$-th centroid. For instance, in the water examples of the manuscript, $q_n = -8e$. The charge neutrality condition implies that the total charge density, i.e., $\\rho_t (r) = \\rho_{\\textrm{ion}} (r) + \\rho_e (r)$, satisfies:\n\\begin{align}\n    \\int dr \\rho_t(r) = \\int dr ( \\rho_{\\textrm{ion}} (r) + \\rho_e (r) ) = 0\n\\end{align}\nThus, the total electrostatic energy is well defined and given by:\n\\begin{align}\\label{eqn:total-e}\n    E_t = \\frac 12 \\int dr dr' \\rho_t(r) v(r-r') \\rho_t(r')\n\\end{align}\nHere $v(r) = 1\/\\vert r\\vert$ is the Coulomb interaction, and it is understood that the (infinite) self-energy of the ionic point charges should be subtracted from the integral. The energy in Eq.~\\eqref{eqn:total-e}  is calculated exactly in the DFT reference model, while it is approximated, in the DPLR model, by the electrostatic energy of a system of spherical Gaussian charge distributions $g_I$ and $g_n$ located at ionic ($R_I$) and WC ($W_n$) sites, respectively. In DPLR one adds to this energy a short-range contribution described by the DP part of the model, which takes care of all the additional many-body effects within $r_c$, the cutoff radius of the model. Thus, all residual errors of the approximation for the electrostatic energy should be associated with long-range effects, for which $r > r_c$.    \nThe Gaussian distributions $g_I$ and $g_n$ integrate to $q_I$ and $q_n$, respectively, but are assumed to have the same spread $1 \/ (\\sqrt 2 \\beta)$, with $\\beta$ an adjustable parameter. \nThe true distributions, $\\rho_I$ and $\\rho_n$, differ from $g_I$ and $g_n$ according to: \n\\begin{align}\\label{eqn:gi-di}\n    &\\rho_I(r) = g_I(r) + \\rho_I(r) - g_I(r) = g_I(r) + \\Delta_I(r)\\\\\\label{eqn:gn-dn}\n    &\\rho_n(r) = g_n(r) + \\rho_n(r) - g_n(r) = g_n(r) + \\Delta_n(r)\n\\end{align}\nThe distributions $\\Delta_I$ and $\\Delta_n$, in Eqs.~\\eqref{eqn:gi-di} and ~\\eqref{eqn:gn-dn}, integrate to zero. $\\Delta_I$ is known, having all its mutipole moments equal to zero, while $\\Delta_n$ has zero dipole moment, but is otherwise unknown. We choose the spread parameter $\\beta$ so that the Gaussian distributions are at the same time smooth on the atomic scale and \nwell localized within $r_c$. More details on the choice of $\\beta$ will be given when discussing specific examples later in the manuscript.\nWhile the electrostatic potential generated by $\\Delta_I$, which has spherical symmetry, decays to 0 exponentially for $r > r_c$, the potential generated by $\\Delta_n$ decays to 0 algebraically for $r > r_c$ because, in general, the distribution $\\Delta_n$ has non-zero quadrupole and higher multipole moments. Thus, the potential generated by $\\Delta_n$ decays at least as fast as $r^{-3}$ for $r > r_c$ and has magnitude controlled by the quadrupole moments specific to the system. Typically, we expect quadrupoles on the order of 1 a.u., which would correspond to $\\Delta_n(r=r_c)$ being approximately equal to 1 meV for $r_c = 6$ A. \nGiven that the DP part of the model takes care of the effects within $r_c$, the error of the above procedure is associated with effects \nbeyond $r_c$, which are controlled primarily by $\\Delta_n$. \n\nThe total ionic and electronic Gaussian distributions are given by:\n\\begin{align}\n    & G_{\\textrm{ion}}(r) = \\sum_I g_I(r-R_I)\\\\\n    & G_e(r) = \\sum_n g_n (r-W_n)\n\\end{align}\nThe  corresponding total Gaussian charge distribution is $G_t(r) = G_{\\textrm{ion}}(r) + G_e(r)$. \nThe electrostatic energy associated with $G_t$ is easily calculated in Fourier space~\\cite{ewald1921die}\n\\begin{align}\\label{eqn:e-dplr}\n    E_{G_t} & = \\frac{1}{2\\pi V} \\sum_{m\\neq 0, \\vert m\\vert \\leq L} \\frac{\\exp(-\\pi^2 m^2\/\\beta^2)}{m^2}  S^2(m),\n\\end{align}\nwhere $L$ is the cutoff in Fourier space, and $S(m)$, the structure factor, is given by:\n\\begin{align}\\label{eqn:sm}\n    S(m) = \\sum_I q_I e^{-2\\pi i m  R_I} + \\sum_n q_n e^{-2\\pi i m W_n}\n\\end{align}\nDue to the smoothness of the distributions, the Fourier sum in Eq.~\\eqref{eqn:e-dplr} converges rapidly. \nFast algorithms for calculating the electrostatic energy are available, such as the smooth particle mesh Ewald (SPME)~\\cite{darden1993pme,essmann1995spm} and the particle-particle-particle-mesh (PPPM)~\\cite{hockney1988computer} methods. The latter is adopted in the applications discussed in this manuscript.\nThe electrostatic energy $E_{G_t}$ in Eq. \\eqref{eqn:e-dplr} contains also short-range contributions. To avoid double counting, $E_{G_t}$ is subtracted from the total potential energy when training the DP part of the DPLR model.  \n\n\nThe error $\\mathcal E$, due to neglecting the contribution of $\\Delta_n$ to the electrostatic energy, is the sum of two terms, $\\mathcal E_1$ and $\\mathcal E_2$, given respectively by:\n\\begin{align}\\label{eqn:error-e1}\n    &\\mathcal E_1 = \\int drdr' G_t(r) v(r-r') \\sum_n\\Delta_n (r') \\\\ \\label{eqn:error-e2}\n    &\\mathcal E_2 = \\int drdr' \\sum_m\\Delta_m(r) v(r-r') \\sum_n\\Delta_n(r)\n\\end{align}\nThe two integrals above are well defined for extended systems as the corresponding charge distributions are neutral.\nSince the lowest non-zero multipole of $\\Delta_n$ is the quadrupole, $\\mathcal E_2$ in Eq.~\\eqref{eqn:error-e2} is (at worst) a sum of quadrupole-quadrupole interactions, decaying as $r^{-5}$. \nOnly the part of that sum involving terms with (roughly) $r>r_c$ cannot be learned by the DP model. \nThese terms contribute to the error. Taking 6~\\AA\\  for a typical $r_c$ value, the sum would include terms smaller than 1~meV.\nDue to the fast rate of decay the double sum should converge rapidly. \nWe could estimate $\\mathcal E_1$ in Eq.~\\eqref{eqn:error-e1} in a similar fashion by writing $G_t$ as a sum of local dipole contributions, resulting in a sum of dipole-quadrupole interactions that decay as $r^{-4}$, i.e., still rapidly but less rapidly than the terms in $\\mathcal E_2$.\nThis estimate is, however, affected by the arbitrariness of the local dipole definition. \nA better way of estimating $\\mathcal E_1$ is the following. Let $E_\\Delta(r)$ be the electric field generated by the sum of the $\\Delta_n$ distributions:\n\\begin{align}\\label{eqn:error-efield}\n    E_\\Delta(r) = \\int dr' \\frac{r - r'}{\\vert r - r'\\vert^3} \\sum_n\\Delta_n(r)\n\\end{align}\nTo leading order, the electric field in Eq.~\\eqref{eqn:error-efield} is a sum of quadrupole contributions that decay like $r^{-4}$ for $r > r_c$. This field is essentially uniform on the molecular scale. Let us call the corresponding average field $\\bar E_\\Delta$. Then, the average energy error can be written in terms of the cell dipole (polarization times volume) $M$ as:\n\\begin{align}\\label{eqn:error-e1-me}\n    \\mathcal E_1 = - M \\cdot \\bar E_\\Delta\n\\end{align}\nIn extended systems $M$ is defined modulo a quantum\\recheck{\\cite{vanderbilt2018berry}}. Thus, the error in Eq. \\eqref{eqn:error-e1-me} is only well defined for a change of ionic coordinates that leads to a change of $M$. \n\nIn summary, the error in the LR contribution to the electrostatic energy, when the true electronic distribution $\\rho_e$ is replaced by \nthe Gaussian distribution $G_e$, is given by a sum of small terms that decay at most as $r^{-4}$ for $r > r_c$. \nIt would be possible to further reduce these errors by representing $G_e$ as a sum of non-spherical Gaussian distributions\nhaving the spreads of the Wannier centroid distributions, which could be learned by an extended DW approach. In this way \nthe lowest non-zero multipole of $\\Delta_n$ would be the octupole. At $r_c$=6~\\AA, such treatment would reduce the error \nby approximately an order of magnitude. However, this formulation would also result in a significant increase of the computational\ncomplexity, which was deemed unnecessary based on our tests.\n\nWe remark that formulations of LR electrostatics based on effective atomic charges cannot reproduce the fluctuations of $M$ \\recheck{in condensed phase}. In these formulations, the atom centered \nelectronic charge distribution differ locally from the true distribution $\\rho_e$ at the dipole rather than at the quadrupole \nlevel. \nTherefore, \nthe corresponding error would be associated with dipole-dipole interactions that are only conditionally convergent. \n\\recheck{It is worth noting that, in finite systems, the error is reduced by adopting partial charge models trained to reproduce the molecular dipole.}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Deep Potential Long-Range model} \n\nIn DPLR the PES is given by\n\\begin{align}\\label{eqn:e-decomp}\n    E = E_\\mathrm{sr} + E_{G_t},\n\\end{align}\nwhere the electrostatic energy $E_{G_t}$ was introduced in Eq.~\\eqref{eqn:e-dplr},\nand $E_\\mathrm{sr}$ is a short-range\ncontribution constructed as in the standard DP model (see Appendix~\\ref{sec:app-srdp} for more details). However, $E_\\mathrm{sr}$\nis not equal to the standard DP energy, because the Gaussian electrostatic energy $E_{G_t}$ is subtracted from the DFT training data to avoid double counting.  \n\nThe forces and the virial are derivatives of the energy with respect to the atomic positions and the simulation cell tensor, respectively.\nWithin DPLR, the energy $E$ depends on the atomic coordinates, both explicitly and implicitly, via the dependence of the WCs on the atomic coordinates. The implicit dependence makes the calculations nontrivial.\n\nThe force on atom $I$ is given by\n\\begin{align}\\label{eq:tmp-a1}\n    F_I = \n    - \\frac{\\partial E}{\\partial R_I} \n    =\n    - \\frac{\\partial E_\\mathrm{sr}}{\\partial R_I} \n    - \\frac{\\partial E_{G_t}}{\\partial R_I} \n    - \\sum_n \\frac{\\partial E_{G_t}}{\\partial W_n} \n    \\frac{\\partial W_n}{\\partial R_I}. \n\\end{align}\nHere the environmental dependence of $W_n$ is given by the DW model. We recall that there is a one-to-one correspondence \nbetween the Wannier centroid $n$ and its nearest atom, the atom $I$. This correspondence establishes a bijective mapping between the indices of the WCs and the indices of a subset of the ions, that we indicate, equivalently, by $n = n(I)$ or by $I = I(n)$. Due to the nearsightedness of the electronic matter~\\cite{}, the position of the WC associated with atom $I$ depends on the local environment\nof that atom within a cutoff radius $r_c$ that can be safely assumed to be equal to the cutoff radius of the DP model. \nIndicating with $D_n$ the displacement of $W_n$ relative to $R_I$, we have: \n\\begin{align}\n    W_n = R_{I(n)} + D_n,\n\\end{align}\nwhich relates the environmental dependence of $D_n$ to that of $W_n$. The latter \nis smooth by construction in the DW model.\nThus,\n\\begin{align}\n    \\frac{\\partial W_n}{\\partial R_J}\n    = \n    \\delta_{I(n),J} + \\frac{\\partial D_n}{\\partial R_J},\n\\end{align}\nwhere $\\delta_{I,J}$ is the Kronecker delta, and the force on atom $I$ in Eq. \\eqref{eq:tmp-a1} becomes\n\\begin{align}\\label{eq:app-f}\n     F_I = \n     - \\frac{\\partial E_\\mathrm{sr}}{\\partial R_I} \n    - \\frac{\\partial E_{G_t}}{\\partial R_I} \n    - \\frac{\\partial E_{G_t}}{\\partial W_{n(I)}} \n    - \\sum_n \\frac{\\partial E_{G_t}}{\\partial W_n} \n    \\frac{\\partial D_n}{\\partial R_I}. \n\\end{align}\nThe first term on the right-hand-side (RHS) of Eq. \\eqref{eq:app-f} is the standard DP force, the second and third terms give \nthe electrostatic force on atom $I$ and its associated WC, respectively, while\nthe fourth term originates from the environmental dependence of the WC.\nThe derivatives $\\partial D_n\/\\partial R_I$ are calculated by back-propagating the DW model.\n\nWe indicate the cell tensor by $h = \\{h_{\\alpha\\beta}\\}$, with $h_{\\alpha\\beta}$ being the $\\beta$-th component of $\\alpha$-th cell vector. \nThe virial is defined by\n\\begin{align}\\label{eq:app-v0}\n\\Xi_{\\alpha\\beta} \n= -\\frac{\\partial E}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta},\n\\end{align}\nwhere summation over repeated indices is assumed.\nBy using Eq.~\\eqref{eqn:e-decomp} we have\n\\begin{align}\\label{eq:app-v1}\n\\Xi_{\\alpha\\beta} = -\\frac{\\partial E_\\mathrm{sr}}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}  -\\frac{\\partial E_{G_t}}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}\n= \\Xi^\\mathrm{sr}_{\\alpha\\beta} -\\frac{\\partial E_{G_t}}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta},  \n\\end{align}\nThe first term on the RHS of Eq.~\\eqref{eq:app-v1} is the short-range virial contribution, $\\Xi^\\mathrm{sr}_{\\alpha\\beta}$,\nwhich is calculated as in the standard DP model. The remaining term is the electrostatic virial contribution, whose calculation is non-trivial as we detail below. \nFirst, it is convenient to define the reciprocal cell tensor $h^{-1} = \\{h^{-1}_{\\beta\\alpha}\\}$, in which $h^{-1}_{\\beta\\alpha}$ is the $\\beta$-th component of $\\alpha$-th reciprocal cell vector. The direct and reciprocal cell tensors define linear scaling transformations in direct and\nreciprocal space that are useful in MD simulations with variable cell.\nIn direct space, a scaled coordinate $s$ is related to its non-scaled counterpart $R$ by $R = sh$, and, in reciprocal space, \nthe lattice vector $m$ is related to its scaled counterpart $k$ by $m = h^{-1}k$.\nAs a consequence, the scalar product \n$mR = sk$ is independent of the cell tensor $h$, and so is the factor $e^{2\\pi imR_I}$ in Eq.~\\eqref{eqn:sm}. However, the factor $e^{2\\pi imW_I}$ in the same equation does depend on the cell vector $h$, because in general \nthe positions of the WCs do not change linearly with a cell deformation.\nThen, we can write for the electrostatic contribution to the virial:\n\\begin{align}\\label{eq:tmp-a4}\n    -\\frac{\\partial E_{G_t}}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}\n    =\n    \\Xi^{\\mathrm{rec}}_{\\alpha\\beta}\n    +\n    \\Xi^{c}_{\\alpha\\beta},\n\\end{align}\nwith\n\\begin{align}\\label{eq:tmp-a5}\n    \\Xi^{\\mathrm{rec}}_{\\alpha\\beta} & = \n    -\\frac{\\partial}{\\partial h_{\\gamma\\alpha}} \\Big(\\frac1{2\\pi V} \\Big)\n    h_{\\gamma\\beta} \\sum_{m\\neq0}f(m)S^2(m)\n    -\n    \\frac1{2\\pi V} \\sum_{m\\neq0}\\frac{\\partial f(m)}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta} S^2(m) \\\\ \\label{eq:tmp-a6}\n    \\Xi^{c}_{\\alpha\\beta} &= \n    -\n    \\frac1{\\pi V} \\sum_{m\\neq0}f(m)\n    S(m) \\frac{\\partial S(m)} {\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}, \n\\end{align}\nwhere we used the definition $f(m) = \\frac1{m^2}{\\exp(-\\pi^2m^2\/\\beta)}$.\n$\\Xi^\\mathrm{rec}_{\\alpha\\beta}$ in Eq.~\\eqref{eq:tmp-a5} is the standard reciprocal virial contribution of the Ewald method, while the correction term in Eq.~\\eqref{eq:tmp-a6} accounts for the nonlinear dependence of the WCs on the cell tensor. \nThe correction term $ \\Xi^{c}_{\\alpha\\beta}$ is calculated as follows\n\\begin{align}\\label{eq:tmp-a8}\n    \\Xi^{c}_{\\alpha\\beta}\n    =\n    -\n    \\frac1{\\pi V} \\sum_{m\\neq0}f(m) S(m)\n    \\sum_n q_n e^{2\\pi iW_n m} 2\\pi i \n    \\Big(\n    \\frac{\\partial W_{n\\delta}}{\\partial h_{\\gamma\\alpha}} m_\\delta +\n    W_{n\\delta} \\frac{\\partial m_\\delta}{\\partial h_{\\gamma\\alpha}}\n    \\Big)\n    h_{\\gamma\\beta}, \n\\end{align}\nwhere the terms in parentheses are given by\n\\begin{align*}\n    \\frac{\\partial W_{n\\delta}}{\\partial h_{\\gamma\\alpha}} m_\\delta h_{\\gamma\\beta}\n    &= \n    \\sum_J \\frac{\\partial W_{n\\delta}}{\\partial R_{J\\epsilon}}\\frac{\\partial R_{J\\epsilon}}{\\partial h_{\\gamma\\alpha}} m_\\delta h_{\\gamma\\beta}\n    =\n    \\sum_J \\frac{\\partial W_{n\\delta}}{\\partial R_{J\\alpha}} R_{J\\beta} m_\\delta\n    = \n    R_{n\\beta} m_\\alpha\n    +\n    \\sum_J \\frac{\\partial D_{n\\delta}}{\\partial R_{J\\alpha}} R_{J\\beta} m_\\delta\\\\\n    W_{n\\delta} \\frac{\\partial m_\\delta}{\\partial h_{\\gamma\\alpha}} h_{\\gamma\\beta}\n    & =\n    W_{n\\delta} \\frac{\\partial h^{-1}_{\\delta\\epsilon}}{\\partial h_{\\gamma\\alpha}} k_\\epsilon h_{\\gamma\\beta}  \n    =\n    -W_{n\\delta}h^{-1}_{\\delta\\eta} \\frac{\\partial h_{\\eta\\zeta}}{\\partial h_{\\gamma\\alpha}} h^{-1}_{\\zeta\\epsilon} k_\\epsilon h_{\\gamma\\beta}\n    =\n    -W_{n\\delta}h^{-1}_{\\delta\\gamma}  h^{-1}_{\\alpha\\epsilon} k_\\epsilon h_{\\gamma\\beta}\n    = - W_{n\\beta} m_\\alpha\n\\end{align*}\nThus, the correction virial term in Eq.~\\eqref{eq:tmp-a8} becomes\n\\begin{align}\n    \\Xi^{c}_{\\alpha\\beta}\n    =\n    -\n    \\frac1{\\pi V} \\sum_{m\\neq0}f(m) S(m)\n    \\sum_n q_n e^{2\\pi iW_n m} 2\\pi i \n    \\Big(\n    D_{n\\beta}m_\\alpha \n    +\n    \\sum_J \\frac{\\partial D_{n\\delta}}{\\partial R_{J\\alpha}} R_{J\\beta} m_\\delta\n    \\Big)\n\\end{align}\nGiven that \n\\begin{align}\n    -\\frac{\\partial E_{G_t}}{\\partial W_{n\\alpha}}\n    =\n    -\n    \\frac1{\\pi V} \\sum_{m\\neq0}f(m) S(m)\n    2\\pi i m_\\alpha q_n e^{2\\pi iW_n m}, \n\\end{align}\nthe correction virial simplifies to\n\\begin{align}\\label{eq:tmp-a11}\n     \\Xi^{c}_{\\alpha\\beta}\n     =\n     - \\sum_n \\frac{\\partial E_{G_t}}{\\partial W_{n\\alpha}} D_{n\\beta} \n     -\n     \\sum_{nJ} \\frac{\\partial E_{G_t}}{\\partial W_{n}}\n     \\frac{\\partial D_{n}}{\\partial R_{J\\alpha}} R_{J\\beta}\n\\end{align}\nFinally, using Eqs.~\\eqref{eq:app-v1}, \\eqref{eq:tmp-a4}, \\eqref{eq:tmp-a5} and  \\eqref{eq:tmp-a11}, the virial takes the more compact form\n\\begin{align}\n    \\Xi_{\\alpha\\beta} = \\Xi^\\mathrm{sr}_{\\alpha\\beta} + \n    \\Xi^\\mathrm{rec}_{\\alpha\\beta}\n    - \\sum_{nJ} \\frac{\\partial E_{G_t}}{\\partial W_{n}}\n     \\frac{\\partial D_{n}}{\\partial R_{J\\alpha}} R_{J\\beta}\n    - \\sum_n \\frac{\\partial E_{G_t}}{\\partial W_{n\\alpha}} D_{n\\beta} \n\\end{align}\n\n\nWe checked that the above formulae are correct by comparing the analytical derivatives in the formulae to the numerical    \nderivatives calculated with finite differences. As a consequence,\nin molecular dynamics simulations the conservation laws associated with the equations of motion should be satisfied within the accuracy of the adopted numerical implementation. \nWe find, for instance, that in NVE trajectories of liquid water with 128 molecules, the total energy shows a small drift of approximately $0.4$~meV\/\\chem{H_2O} ($\\sim$1~K) per 100 ps with the DPLR approach, when using a mesh spacing $\\eta=0.98$~\\AA\\ in the calculation of the Ewald sum. As shown in Fig.~\\ref{fig:ener_drift}, this is small but greater than the drift of a DP trajectory for the same system, which is almost not observable in a 100~ps trajectory.\nThe total energy drift in the DPLR trajectory is controlled by the numerical accuracy of the fast algorithm for computing the Ewald sum, here the PPPM method, and can be further reduced by using a finer mesh.\nAs shown in Fig.~\\ref{fig:ener_drift}, the drift of DPLR is not observable when the mesh spacing $\\eta$ is set equal to $0.49$~\\AA.\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{fig-ener-drift.pdf}\n    \\caption{Drift of the total energy of DP and DPLR models in a 100~ps NVE simulation of 128 water molecules. \n    The time-step is set to 0.5~fs.\n    Two mesh spacings of the PPPM method, i.e.~$\\eta = 0.98$~\\AA\\ and $\\eta = 0.49$~\\AA, are used in the DPLR model.\n    }\n    \\label{fig:ener_drift}\n\\end{figure}\n\n\n\\section{Deep Potential Long-Range model for water}\n\nWe construct a DPLR model for water based on the PBE functional approximation of DFT~\\cite{perdew1996generalized}. \nIt is well known that PBE substantially overestimates the strength of the hydrogen-bonds in water. \nAs a consequence, it fails to describe correctly some important thermodynamic properties, like the relative density of ice and water at ambient pressure~\\cite{gaiduk2015density}.\nHowever, here we are not interested in constructing a state-of-the-art water model, but rather we intend to demonstrate that DPLR can capture long-range electrostatic effects missing in the DP model.\n\\recheck{For this purpose, the choice of the exchange-correlation functional is not a critical issue. It would be straightforward to adopt more accurate functionals than PBE by simply replacing PBE with any other functional of choice, in the labeling steps whereby energy, force and virial tensor are calculated at selected atomic configurations.\n}\nIn the next three subsections, we construct DP, DW, and DPLR models based on the PBE functional approximation.  \n\n\n\\subsection{Training data and DP model}\n\nWe use the concurrent learning scheme Deep Potential GENerator (DP-GEN)~\\cite{zhang2019active,zhang2020dp}. DP-GEN iteratively enlarges the training dataset of labeled DFT data and refines a representative ensemble of DP models by learning from a growing dataset. In this work the representative ensemble consists of four models that differ in the random initialization of the network parameters but are otherwise trained on the same data. In the DP-GEN protocol \nthe four DP models are used to sample the relevant thermodynamic space with deep potential molecular dynamics (DPMD) trajectories. The DP construction includes three stages.\nIn the first one (iteration 1 to 7), bulk configurations in the thermodynamics space $200 \\leq T \\leq 400$~K and $1\\leq P \\leq 10^4$~Bar are explored with isothermal-isobaric (NPT) DPMD. \nThe initial configurations for bulk NPT DPMD simulations are snapshots of DPMD simulations with 128 molecules in the liquid state obtained with the DP model developed in Ref.~\\citen{zhang2018deep}.\nThe second stage extends from iteration 8 to iteration 15. In this stage slab configurations are explored with canonical (NVT) DPMD simulations in the temperature range $200 \\leq T \\leq 400$~K.\nThe initial configurations are prepared by adding a vacuum region on top of bulk liquid configurations. \nDifferent surface structures with thickness of the vacuum region ranging from 0.5~\\AA\\ to  7.0~\\AA\\ are considered.\nIn the third stage, from iteration 16 to 23, low density bulk configurations are explored by NVT simulations for $200 \\leq T \\leq 400$~K. Initial configurations are created by randomly removing 64, 96, 112 and 120 molecules from the initial configurations of stage 1.\nAt each stage, the time span of the DPMD simulations is gradually increased over the iterations, from 1~ps to 10~ps.  \nThe maximal deviation of the forces predicted by the ensemble of DP models is monitored along the trajectories and used to classify the explored configurations. \nConfigurations with deviation between 0.15 to 0.30~eV\/\\AA\\ are labelled as training data, following the protocol detailed in Ref.~\\citen{zhang2020dp}.\nSingle-shot DFT calculations are performed at these configurations.\n\nDFT calculations are carried out with the Vienna {\\it ab initio} simulation package (VASP) version 5.4.4~\\cite{kresse1996efficiency,kresse1996efficient},\nusing the PBE exchange-correlation energy. \nThe kinetic energy cut-off for the planewave basis is set to 1500~eV. \nThe self-consistent field procedure is stopped when the energy difference between the last two iterations is less than $10^{-6}$~eV.  \n\nThe DP models are trained with the DeePMD-kit package~\\cite{wang2018deepmd}. \nThe cut-off radius $r_c$ for the atomic environments is set to 6~\\AA. \nThe size of the embedding and that of the fitting network are set to (25, 50, 100) and (240, 240, 240), respectively. \nThe DP models are trained with the Adam stochastic gradient descent scheme~\\cite{kingma2015} using $10^6$ steps, during which the learning rate exponentially decays from $10^{-3}$ to $3.5\\times 10^{-8}$.\nAt each training step, a subset of the training dataset, referred to as the mini-batch, is used to calculate the gradient of the loss function. \nThe size of the minibatch is 1, i.e.~only 1 configuration is used for calculating the gradient.\n\nAt the end of each training stage the fraction of configurations with gradient deviation larger than 0.15~eV\/\\AA\\ is reduced to less than 0.1\\%, indicating satisfactory convergence of the DP-GEN cycle.\nA total of 448 configurations are selected as training data during the DP-GEN protocol.\nThey include 323 bulk,  94 surface, and 31 low density configurations.\nThe dataset used for training the initial DP models has 135 configurations, thus the total training data includes 583 configurations. \n\n\n\\subsection{DW model}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{fig-accuracy_corr.pdf}\n    \\caption{Training accuracy of the DW model for water. The three panels report the three Cartesian displacements of the Wannier centroids (WC) relative to their nearest oxygen atom predicted by DW (vertical axis) and by DFT (horizontal axis).}\n    \\label{fig:cwc-train-err}\n\\end{figure}\n\nFor each water configuration, the DW model gives the position of the WC associated with a given molecule relative to the O atom of that molecule. \nThe training data are the same data used to generate the DP model.\nEach WC is uniquely associated with a particular O atom and depends on the atomic environment of that atom within a cutoff radius, which is set here to 6~\\AA, i.e., it is the same cutoff $r_c$ of the DP model. \nWe use the standard DP descriptor $\\mathcal D = \\mathcal D^a$ (see Appendix for details).\nThe sizes of the embedding and fitting networks are (25, 50, 100) and (100, 100, 100), respectively. \nThe model is trained with the Adam stochastic gradient descent method~\\cite{kingma2015} using $10^6$ steps. \nThe batch size is set equal to 1. The learning rate exponentially decays from $1.0 \\times 10^{-2}$ to $5.6\\times 10^{-8}$.\n\nThe training accuracy of the DW model is $1.7\\times 10^{-3}$~\\AA, which is much smaller than the typical distance $D$ of a WC from its reference O atom. \nThe test accuracy is $1.9\\times 10^{-3}$~\\AA. \nThus, the generalization gap between test and training errors is almost negligible.\nThe correlation between labelled and predicted $D$ is graphically illustrated in  Fig.~\\ref{fig:cwc-train-err}.\n\n\n\\subsection{DPLR model}\\label{sec:lrdp-train}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig-error-rc4-6-e.pdf}\n    \\caption{Root mean square error (RMSE) of the energy predicted by the DPLR model as a function of the spread parameter $\\beta$. The training error of bulk (solid squares connected by a red solid line) and surface configurations (solid circles connected by a red dashed line), and the test error of surface configurations (open circles connected by a red dotted line) are shown (see text for more details).\n    The corresponding errors for the DP model are plotted as a black solid line (bulk training error), a black dashed line (surface training error), and a black dotted line (test error).\n   \n    }\n    \\label{fig:error}\n\\end{figure}\n\nThe DPLR model is trained on the dataset generated by DP-GEN.\nIn DPLR, the spread parameter $\\beta$, i.e.~the inverse spread of the Gaussian charge distribution, needs to be fixed. \nIn the limit of $\\beta$ going to 0, the Gaussian width is infinite and DPLR reduces to the standard DP model.\nOn the other hand, for $\\beta$ going to infinity, the Gaussian charges become point charges. \nIn this limit, the magnitude of the intramolecular Coulomb force between the oxygen ion ($q_I=6e$) and its associated WC ($q_n=-8e$), for a typical separation distance ($D\\approx 0.07$\\AA), is about $10^4$~eV\/\\AA\\ , which is about four orders of magnitude larger than the average atomic force. \nThis situation would create serious numerical problems due to the difficulty of resolving a labeled force from the strong intramolecular Coulomb force.\nTherefore, there should be an optimal $\\beta$, neither too small to avoid reduction to the standard DP model, nor too large to avoid distributions too close to point-charges. \n\nTo study the effect of the spread parameter on the accuracy of the DPLR models, we train different DPLR models with\ndifferent spread parameters ($\\beta = \\{0.1, 0.2,\\dots,0.6\\}$\\AA$^{-1}$). \nTo ensure proper convergence of the Ewald sum in Eq.~\\eqref{eqn:e-dplr}, the mesh spacing $\\eta$ is set equal to the largest value compatible with the constraint $\\eta\\cdot\\beta \\leq 0.4$.  \nIn DP and in the short-range part of the DPLR models we adopt a hybrid descriptor (see Appendix for details) $\\mathcal D = (\\mathcal D^a, \\mathcal D^r)$, and the cut-off of $\\mathcal D^a$ is set to $r_c^a = 4$~\\AA, while the cut-off of $\\mathcal D^r$ is set to $r_c = 6$~\\AA. The size of the embedding net of  $\\mathcal D^a$ is (25, 50, 100), and that of  $\\mathcal D^r$ is (10, 20, 40).\nTraining and test errors, as a function of $\\beta$, are shown in Fig.~\\ref{fig:error}.\nThe training dataset is divided into two parts, one consisting only of bulk configurations and the other consisting only of surface configurations. \nThe test dataset includes surface configurations with 128 molecules with a vacuum region of 7 or 12~\\AA\\ , generated along a 100-ps DPLR molecular dynamics  \n(DPLRMD) trajectory for each of the two widths of the vacuum region. Configurations are recorded every 10~ps, and are labeled with \\emph{ab initio} DFT calculations. Thus, the test dataset includes 20 surface configurations in total. \n\nIn Fig.~\\ref{fig:error}, one can see that the energy training error for the bulk configurations is smaller than the corresponding error for the surface configurations in both DP and DPLR models, reflecting \nthe more difficult task of interpolating the PES for surface than for bulk configurations, due to the larger variance of the local environments in the former case. For $\\beta < 0.5$~\\AA$^{-1}$, the DPLR training errors are close to those of DP, but for $\\beta \\geq 0.5$~\\AA$^{-1}$ they gradually\nincrease and become notably larger than the corresponding DP errors, a behavior that we attribute to numerical difficulties with\nGaussian charges that are too localized.\n\nThe test dataset is composed of independent configurations not included in the training dataset, and serves to assess the capability of the\nthe models to deal with unforeseen situations. \nIn the case of the standard DP model, the test error (2.0~meV\/\\chem{H_2O}) is larger than the surface training error (1.2~meV\/\\chem{H_2O}). \nThe gap, usually called generalization gap, between test and training errors indicates that a model overfits the training data and looses accuracy when generalized to cases not included in the training dataset. \nThe DPLR model with a small $\\beta$ has a generalization gap comparable to that of the standard DP model, as expected because for $\\beta \\rightarrow 0$~\\AA$^{-1}$ DPLR reduces to DP.\nHowever, for $\\beta \\approx 0.4$~\\AA$^{-1}$, the generalization gap is minimal. In this case \nthe training error and the test error are 1.3 and 1.5~meV\/\\chem{H_2O}, respectively,\nindicating that inclusion of long-range electrostatics contributes to the model's generalization ability.\nFor $\\beta \\geq 0.5$~\\AA$^{-1}$ the test error increases with $\\beta$, as expected from numerical difficulties \nwith charge distributions that are too strongly localized.\n\n\nBased on the above analysis, the optimal spread parameter $\\beta$ should have a value of about 0.4~\\AA$^{-1}$. It is interesting to note that, with this value of the spread parameter, the radial extent of the Gaussian distribution $g_n(r)$ about the oxygen atom is close to the physical extent of the (valence) electron density of a water molecule, as measured by the radius of the surface on which the electron density is approximately one order of magnitude smaller than its maximum value. Thus, with the optimal $\\beta$ the spherical Gaussian distribution $g_n(r)$ approximates (roughly) the physical WC distribution $\\rho_n(r)$ and the error $\\Delta_n(r)$ in Eq.~\\eqref{eqn:gn-dn} is minimized.  \n\nIn the rest of the manuscript, we set $\\beta = 0.4$~\\AA${}^{-1}$ , unless stated otherwise.\nWe note that simply increasing the cut-off radius of the standard DP model to 8~\\AA\\ only marginally reduces training and test errors. Indeed, when using $r_c=8$~\\AA\\ in place of $r_c=6$~\\AA,  \nthe bulk and surface training errors are reduced from 1.02~meV\/\\chem{H_2O} and 1.26~meV\/\\chem{H_2O} to 1.01~meV\/\\chem{H_2O} and 1.18~meV\/\\chem{H_2O}, respectively, while the\ntest error is reduced from 2.14~meV\/\\chem{H_2O} to 2.08~meV\/\\chem{H_2O}.\nThis indicates that the long-range effect is non-trivial and can only be captured by including explicitly \nthe long-range electrostatic interaction.\n\n\n\n\n\\section{Application to two water systems}\n\nWe compare the performance of DPLR and DP in two test cases. In one we compute the potential energy of interaction of two water molecules \nas a function of the relative distance. \nIn the other we compute the free energy profile a water molecule versus its distance from a liquid water slab. \nThe second example requires the calculation of a thermal property of a relatively complex system. \nIn both cases, long-range electrostatic interactions\namong the electric dipoles of the water molecules play a role. \nThese interactions, absent in the DP model, are described with sufficient accuracy by DPLR. \n\n\n\\subsection{Water dimer}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{fig-dimer-aligned-1.pdf}\n    \\caption{Potential energy of the water dimer as a function of the separation distance between the two water molecules. \n    The standard deviation of four independently trained DPLR models with the same spread parameter $\\beta = 0.4$~\\AA$^{-1}$ is shown by the error bars in the plot. The top panel compares DFT with DP and DPLR models with cutoff radii $r_c^a = r_c=4$\\AA.\n    \\recheck{The bottom panel compares DFT with DP and DPLR models with cutoff radii $r_c^a = 4$~\\AA\\ and $r_c=6$\\AA .\n    All energies are referred to the potential energy minimum $E_{min}$.}\n   \n    }\n    \\label{fig:dimer}\n\\end{figure}\n\nThe potential energy curve of a water dimer as a function of the separation distance $d_O$ between the two molecules is reported in Fig.~\\ref{fig:dimer}. \n\\recheck{Both DP and DPLR models describe equally well the potential energy at short distance.\nThe DP model does not correctly predict the energy curve at a distance beyond the cut-off radius. \nDue to the finite range of the DP model, the corresponding potential energy curve misses the\n$1\/d_O^3$ tail of the DFT reference, which is due to dipole-dipole interactions. \nBy contrast, the long-range Coulomb tail is recovered accurately by the DPLR model, as shown in both panels of Fig.~\\ref{fig:dimer}.\nThe accuracy at intermediate distance improves by increasing the cutoff in all the models, as expected. \nAlthough DPLR has a higher ability of representing the energy surface, it is marginally less accurate than the DP model between 3.5 and 4.0~\\AA. This can be explained by the fact that dimer configurations were not included in the training dataset.\nFrom Fig.~\\ref{fig:dimer} one concludes that DPLR is a superior model with both cutoffs.\n}\n\n\n\n\n\n\n\\subsection{Free energy profile of a molecule at varying distance from a liquid slab}\n\n\nIn this subsection, we test the performance of DPLR in a calculation of the\nfree energy profile of a water molecule as a function of its distance from a liquid water slab. The model has cut-off radii of $r_c^a = 4$~\\AA\\ and $r_c = 6$~\\AA, spread parameter of 0.4~\\AA$^{-1}$, and Ewald mesh spacing of 1~\\AA. We simulate the slab and the adjacent vacuum region on a periodic box of size 50\\AA$\\times$ 11\\AA $\\times$ 11\\AA. The slab contains 64 molecules, let to equilibrate with a long canonical MD run at a temperature of 300~K starting from an initial bulk liquid configuration. After equilibration, the half width of the slab is $\\sim 7.9$~\\AA ~and the adjacent vacuum region has a width of $\\sim 34.2$~\\AA.\nWe insert an additional molecule to this system, initially near the center of the vacuum region where it interacts weakly with the slab. Then, we vary the distance of the molecule from the slab and we calculate the free energy profile as a function of the distance, while keeping the whole system in equilibrium at 300~K. The distance of the molecule from the slab is defined by the distance of the respective centers of mass. A typical configuration is illustrated in the inset of Fig.~\\ref{fig:fe} for a distance $s$ between the molecule and the liquid slab. In the atomic configuration $\\{R\\}$, the distance of the molecule from the slab is represented by the collective variable $\\sigma(\\{R\\})$. In MD simulations an holonomic constraint, such as $\\sigma(\\{R\\}) = s$, is easily imposed by a Lagrange multiplier. We set the initial distance to $s_{\\textrm{far}} = 17$~\\AA. At this distance the molecule experiences weak long-range dipolar interactions with the slab. We indicate by $F$ the projection of the force on the molecule along the direction connecting the center of mass of the molecule and that of the slab. When the molecule is displaced from $s_{\\textrm{far}}$ to $s$, the corresponding free energy change, $A(s)$, is given by: \n\\begin{align}\\label{eq:pomf}\n    A(s) = \\int_s^{s_{\\textrm{far}}} \\langle F \\rangle^\\tau d \\tau,\n    \\quad\n    \\langle F \\rangle^\\tau = \\frac1Z \\int F e^{-\\beta H(\\{R\\})} \\delta(\\sigma(\\{R\\}) - \\tau) d\\{R\\}.\n\\end{align}\nHere $H(\\{R\\})$ and $Z$ are the Hamiltonian and the partition function of the system. \nFor each value of $\\tau$ the ensemble average in Eq. \\eqref{eq:pomf} is calculated with canonical MD at 300~K for configurations that satisfy the constraint $\\sigma(\\{R\\}) = \\tau$. We compute the ensemble average at a discrete set of $\\tau$ values using 1000~ps long MD trajectories, the first 200 ps of which serve for equilibration and are not included in the observation time. During observation configurations are recorded every 10~ps.\nBy varying $s$ from $s_{\\textrm{far}}$ to $s_{\\textrm{near}} = 9$~\\AA, we obtain the blue free energy profile reported in panel (a) of Fig.~\\ref{fig:fe} for the DPLR model. We divide the interval $[s_{\\textrm{near}}, s_{\\textrm{far}}]$ into 8 equal intervals and calculate the corresponding free energy changes by\nevaluating the integrals in Eq. \\eqref{eq:pomf} numerically with the trapezoidal rule.   \nFor $s$ in the vicinity of $s_{\\textrm{near}}$ the tagged molecule forms hydrogen bonds with neighboring molecules in the slab, suggesting that incorporation into the slab is taking place. \n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig-fe.pdf}\n    \\caption{The free energy of water molecule absorption to a water slab.\n    The panel (a) presents the free energy calculated by DFT, DPLR and DP models. \n    The panel (b) presents the error in free energy of the DPLR and DP models with respect to DFT.\n    The error bar in the plot is given by the standard deviation of four independently trained models.\n    The insert is a schematic plot of the system configuration.\n    }\n    \\label{fig:fe}\n\\end{figure}\n\n\nWe use a simple approximation to estimate the deviation of the calculated DPLR profile from the DFT reference. Instead of re-weighting with the DFT energy the configurations along DPLR trajectories, which would be expensive, given the large number of configurations needed for statistical accuracy, we keep the DPLR weights and approximate the ensemble average with the following expression, which only requires DFT calculations at a relatively small number of configurations:\n\\begin{align}\\label{eq:approx}\n    \\langle F_\\mathrm{DFT} \\rangle^\\tau_\\mathrm{DFT} \\approx \\langle F_\\mathrm{DFT} \\rangle^\\tau_\\mathrm{DPLR}.\n\\end{align}\nThis procedure can be justified because DPLR reproduces accurately DFT energies with an average error of $\\sim 1$~meV\/\\chem{H_2O}. \nWe find that 80 configurations, equally spaced in time along an equilibrated 800~ps long trajectory, are sufficient to compute the average force at each $\\tau$ value. \nThe results are reported in panel (b) of Fig.~\\ref{fig:fe}, where the baseline DFT reference corresponds to $\\Delta A(s) = 0$. \nThe deviations of DPLR from the baseline, calculated at discrete distances in the interval $[s_{\\textrm{near}}, s_{\\textrm{far}}]$, appear in the same figure as blue points connected by blue lines. \nThe blue points represent the average of four independent DPLR models trained on the same data with same hyper-parameters but different random initialization of the model parameters.\nThe blue error bars are the corresponding standard deviations of the predictions of the four models. Overall, the deviation of DPLR from DFT is quite small and the uncertainty of the model, estimated from the standard deviation of the four models, is even smaller.     \n\nWe now consider a DP model with same cutoff radius of DPLR but without explicit long-range Coulomb interactions. To estimate the deviation of this model from DPLR  we adopt a similar approximation to the one adopted in Eq.~\\eqref{eq:approx} , i.e., we assume:\n\\begin{align}\n    \\langle F_{\\textrm{DP}} \\rangle^\\tau_{\\textrm{DP}} \\approx \\langle F_{\\textrm{DP}}\\rangle^\\tau_\\mathrm{DPLR},\n\\end{align}\nand use the same configurations as before to calculate the ensemble averages and the free energy differences. The deviations of DP from DPLR and from the baseline DFT reference are reported as red points with error bars connected by red lines in panel (b) of Fig.~\\ref{fig:fe}. As before, the DP points are averages of four independent models trained on the same data with different random initialization. The average free energy profile obtained in this way is reported as a red line in panel (a) of the same figure, showing that when the molecule approaches the slab the gain in free energy is smaller for DP than for DPLR, a result consistent with the missing attractive dipolar interactions in the DP model. However, the most important outcome of this analysis is that the DP model is affected by a large uncertainty. When evaluated along DPLR trajectories the standard deviation of the independent DP models grows rapidly as the tagged molecule approaches the slab to become almost an order of magnitude larger than the uncertainty of the corresponding DPLR model when the tagged molecule is near the slab. We attribute this behavior to the absence of long-range electrostatic interactions in the DP model as opposed to DPLR and DFT.        \n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Phonons in crystalline sodium chloride}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{fig-gamma-all.pdf}\n    \\caption{The LO-TO splitting of the phonon spectra of NaCl crystal.\n    DPLR, DP, DFT and experiment (Expt.)~\\cite{raunio1969phonon} are compared. \n    (a) Spectra obtained from the analytical dynamical matrix (see text) on a $2\\times2\\times2$ (64 atoms) supercell.\n    (b) Spectra obtained from the full dynamical matrix, including  analytical and non-analytical correction (NAC) terms (see text), on a $2\\times2\\times2$ supercell.\n    (c-e) Spectra obtained from the analytical dynamical matrix on a $3\\times3\\times3$ (216 atoms) (c), $5\\times5\\times5$ (1000 atoms) (d), and $6\\times6\\times6$ (1728 atoms) (e) supercell, respectively.\n    }\n    \\label{fig:nacl}\n\\end{figure}\n\nIn this section we consider the phonon dispersion curves in the sodium-chloride (NaCl) crystal, as a further example to illustrate the importance of long-range electrostatic interactions. \nFor the reference DFT model we adopt the PBE exchange-correlation functional and use pseudopotentials for the electron-ion interactions. DP and DPLR models for NaCl are trained on DFT data generated in a 200~ps DPMD NPT trajectory of a $2\\times 2\\times 2$ supercell (64 atoms) at 100~K and 1~Bar. In the DFT data we use  Brillouin Zone (BZ) sampling with  \na $4\\times 4\\times 4 $ Monkhorst Pack mesh to calculate the interatomic potential and the maximally localized Wannier distributions. The hyper-parameters of the DP and DPLR models are the same as in the water models discussed earlier in the manuscript. Using the same arguments of \nSect.~\\ref{sec:lrdp-train}, we find that the optimal value of the spread parameter for NaCl is $\\beta = 0.2$~\\AA$^{-1}$. The smaller $\\beta$ of NaCl compared to water reflects the fact that the electron density distribution about the Cl atom in NaCl is more delocalized than the one about the O atom in the water molecule. The training accuracy of the DP model is $7.6\\times 10^{-4}$~eV\/atom for energy and $2.0\\times 10^{-3}$~eV\/\\AA\\ for atomic forces,  similar to that of  the DPLR model, which is $7.7\\times 10^{-4}$~eV\/atom for energy and $2.6\\times 10^{-3}$~eV\/\\AA\\ for atomic forces. \n\n\n\nTo compute the phonons, we calculate numerically the force constants from the second derivatives of the PES with respect to atomic displacements. We then obtain the phonon frequencies at a generic point $\\bm q$ of the BZ of the crystal with the \\texttt{phonopy} package~\\cite{phonopy}, by diagonalizing the dynamical matrix given by a Fourier expansion of the force constants at wavevector $\\bm q$. This approach would be adequate in non-polar materials where the force constants have finite range. In polar materials, like NaCl, long-range electrostatic interactions yield a non-analytic contribution to the dynamical matrix, which should be added to the analytic contribution calculated with the above procedure. We report in panel (a) of Fig.~\\ref{fig:nacl} the phonon dispersion curves along the $\\Gamma$-$X$ segment of the BZ obtained from the analytical part of the dynamical matrix by using a $2\\times 2\\times 2$ supercell for the force constants. DP and DPLR curves coincide within the accuracy of the calculation, and are very close to the corresponding DFT curves, as one could have expected given that a $2\\times 2\\times 2$ supercell was used for training the models. Experimental frequencies are also reported in the same panel, from which we see that, while the calculated acoustic and transverse optical (TO) modes agree with experiment, the calculated longitudinal optical (LO) modes deviate considerably from experiment, particularly in the long wavelength limit, where the calculated modes fail to exhibit an LO-TO splitting. Adding the non-analytic contribution to the dynamical matrix restores agreement between theory and experiment, as illustrated in panel (b) of Fig.~\\ref{fig:nacl}. The non-analytic contribution to the dynamical matrix is given by~\\cite{baroni2001phonons}:\n\\begin{align}\\label{eqn:nac}\n\\begin{aligned}\n     \\tilde{D}^{\\textrm{NA}}_{\\kappa\\alpha,\\kappa'\\beta}(\\bm q)\n    &=\n    \\frac{4\\pi}{\\Omega \\sqrt{M_\\kappa M_{\\kappa'}}}\n    \\frac{ \n    (\\sum_\\gamma q_\\gamma Z^\\ast_{\\kappa,\\gamma\\alpha})\n    (\\sum_{\\gamma'} q_{\\gamma'} Z^\\ast_{\\kappa',\\gamma'\\beta})\n    }{\n    \\sum_{\\alpha\\beta} q_\\alpha \\epsilon_{\\alpha\\beta}^\\infty q_\\beta\n    }\n\\end{aligned}\n\\end{align}\nHere, $\\Omega$ is the volume, $\\kappa,\\kappa'$ are atom indices, $\\alpha,\\beta,\\gamma$ are Cartesian directions, ${M_\\kappa},{M_{\\kappa'}}$ are atomic masses, $Z^\\ast_\\kappa,Z^\\ast_{\\kappa'}$ are Born dynamical charge tensors, and $\\epsilon^\\infty$ is the dielectric tensor that describes the static response of the electrons at fixed ions. We recall that the dynamical Born charges are the derivatives of the polarization with respect to atomic displacements, and are therefore directly accessible from the DW network within DPLR. The dielectric tensor is related to the dielectric susceptibility or polarizability tensor $\\chi$ via $\\epsilon^\\infty = 1+4\\pi\\chi$. The susceptibility is defined by the derivatives of the polarization with respect to an applied electric field and is accessible within a DW extension introduced to model the polarizability surface~\\cite{sommers2020raman}. Thus, all the quantities needed for calculating the phonons in polar materials are accessible within the extended DP methodology. \n\nIt is interesting to monitor the evolution of the analytical phonon modes calculated within DP and DPLR when the size of the supercell used to compute the force constants is increased. This is shown in panel (c), (d), and (e) of Fig.~\\ref{fig:nacl} for a $3\\times3\\times3$ supercell (216 atoms), a $5\\times5\\times5$ supercell (1000 atoms), and a $6\\times6\\times6$ supercell (1728 atoms), respectively. The DP modes are essentially independent of size, indicating that, with the chosen cutoff radius ($r_c=6$~\\AA), the force constants for atomic distances larger than those probed in a $2\\times2\\times2$ supercell are negligible. By contrast, the LO mode of DPLR is strongly size dependent, reflecting the appearance of not negligible force constants on larger supercells due to long-range dipolar interactions. With larger supercells the analytical spectra from DPLR recover a larger portion of the correct LO modes. However, the convergence is slow as the non-analytic behavior for $\\bm q \\rightarrow 0$ cannot be recovered from numerical Fourier interpolation. At small wavevectors we observe an upward bump in the LO and TO modes that becomes sharper and moves to smaller $\\bm q$ values as the size of the supercell increases. The bump is a manifestation of the Gibbs oscillations that are expected to occur in place of the LO-TO splitting when attempting to reproduce the non-analytic behavior of the dynamical matrix with a Fourier sum~\\cite{gonze1994interatomic}.\n\nThe long range contributions that are responsible for the spectral changes in panels (c), (d), and (e) derive from the Ewald contribution to the PES within DPLR. This example illustrates an essential advantage of DPLR over DP in materials where long-range electrostatic effects are important. While the DP model can describe well such materials on the supercell used for training, it is unable to extrapolate the PES to larger supercells, a drawback that is absent in the DPLR model.   \n\n\n\n\n\n\\section{External fields}\n\nIn this manuscript we have considered the electrostatic fields originating from internal charges, but, in the linear regime, it is straightforward to include the effect of an external time dependent field $\\mathcal E_\\textrm{ext}(t)$ that couples with the polarization $P(\\{R\\})$. \nIn this situation the PES takes the form:\n\\begin{align}\\label{eq:concl-1}\n    E(\\{R\\}) = \n    E_{\\textrm{DPLR}\/\\textrm{DP}}(\\{R\\}) -\n    P(\\{R\\}) \\cdot \\mathcal E_\\textrm{ext}(t)\n\\end{align}\nIn Eq.~\\eqref{eq:concl-1} the PES in absence of external field can be provided either by DPLR or by DP, depending on whether the size dependence of the electrostatic energy should be included explicitly or not. \nEq.~\\eqref{eq:concl-1} makes possible non-equilibrium MD simulations of an insulating system driven by an external electric field. The response of the system to the external field can also be studied with equilibrium MD using the Kubo formalism, as it was done in Ref.~\\citen{zhang2020deep} to study the infrared absorption spectra of liquid water. \nIn that work the standard DP model for the PES was used, which was adequate because at the infrared frequencies long-range electrostatic effects can be ignored as the dominant correlations are between neighboring molecular dipoles~\\cite{chen2008role}. Long-range correlations among the molecular dipoles become important in the static ionic limit~\\cite{ballenegger2004structure}.\nIn that limit, which is relevant to study the static dielectric constant of water, one should use the DPLR model for the PES in Eq.~\\eqref{eq:concl-1}.              \n \nEq.~\\eqref{eq:concl-1} can be further extended by including the quadratic response of the electrons via the susceptibility tensor $\\chi$:\n\\begin{align}\n    E(\\{R\\}) = \n    E_{\\textrm{DPLR}\/\\textrm{DP}}(\\{R\\}) - \n    P(\\{R\\}) \\cdot \\mathcal E_\\textrm{ext}(t) + \n    \\frac12\\mathcal E_\\textrm{ext}(t) \\cdot \\chi(\\{R\\}) \\cdot \\mathcal E_\\textrm{ext}(t)\n\\end{align}\nThis approach was used in the context of the Kubo formalism to compute the Raman spectra of water in Ref.~\\citen{sommers2020raman}, a study that required the environmental dependence of the electric susceptibility.\nThe latter was modeled by a DNN extending the DW approach to the polarizability tensor. \n\\recheck{The polarizability tensor is defined by the derivatives (i.e. the response) of the Wannier centroid positions with respect to an applied electric field in the limit of zero field. These derivatives are described by an extension of the DW model trained on DFT calculations in presence of a small, i.e. numerically infinitesimal, electric field, as discussed in Ref.~\\citen{sommers2020raman}.}\nThe Raman scattering response is dominated by short range correlations between molecular polarizabilities, and the standard DP model was sufficient for the PES.\n \n\n\\section{Conclusions}\n\nIn this manuscript we examined a fundamental limitation of the local representation of the PES common to most ML models. These models are trained on quantum mechanical data, typically DFT data, on relatively small systems. They are then used in molecular simulations on larger systems that become accessible in view of the computational efficiency of the models. While successful in many cases, this procedure may lead to errors for physical properties affected by long-range electrostatic interactions. We have shown that these effects can be modeled accurately by the DPLR model, which extends the DP methodology using information on the centers of the electronic charge. The latter is encoded in the location of the Wannier centroids, which are averages of the Wannier centers uniquely asssociated to specific atoms. The environmental dependence of the Wannier centroids is described by DW, a DNN model, which we introduced in previous work to describe the dielectric polarization of insulators. Using the information from DW, DPLR augments the local representation of the PES of the standard DP model with the long-range electrostatic energy of ions and electrons, modeled by spherical Gaussian charge distributions centered at ions and Wannier centroids, respectively. The width of the Gaussian distributions is fixed by the value of the spread parameter $\\beta$. Interestingly, the optimal $\\beta$ that minimizes the generalization gap in the ML procedure, is close to the physical value expected from the spatial delocalization of the reference Wannier centroid distribution within DFT, as illustrated in our water and NaCl examples. The variation with size of the long-range electrostatic contribution, ignored in the standard DP model, is approximated accurately in the DPLR model, under the assumption that the electrostatic fields from which this contribution originates are a weak perturbation that can be treated within linear response theory. Because of that, it is possible to learn the environmental dependence of the Wannier centroids ignoring the size dependence of the electrostatic energy as done in the DW scheme.         \n\n\nDPLR uses two DNN models to represent the PES: the standard DP network for the short-range interactions and the DW network for computing long-range electrostatic interactions with the Ewald method.\nBecause of the complex architecture of the DPLR network, whereby force calculations require backward propagation within both DP and DW, and because of the need for Ewald calculations, DPLR simulations are more expensive than standard DP calculations.\nBased on our experience, an increase of the computational burden of \napproximately a factor of 5 should be expected\nwhen using DPLR in place of DP. The decision on which of the two models should be used in a given application should be guided by the physical properties of interest, whether long-range correlations originating by Coulomb forces are important or not. \n\nIn developing the DPLR model we assumed that long-range electrostatic effects beyond the cutoff radius of the DP model originate a weak dependence on size that can be described within the linear response regime. This seems a reasonable assumption based on our experience and on the examples discussed in this manuscript. We cannot exclude, however, that effects of long-range electrostatics beyond the linear response regime may manifest in some circumstances. In these cases, the assumption made here that the environmental dependence of the Wannier centroids is not affected by size should be revisited, and one should envision a self-consistent condition for the centroids under the action of the long-range electrostatic field along lines similar to those developed in Ref.~\\citen{gao2021self}. We expect that a better understanding of all these issues should emerge from widespread application of the DPLR model.\n\n\\recheck{\nThe major simplifying assumption of the current DPLR model is that the WCs are uniquely associated to specific atoms. Thus, they cannot split or recombine along molecular dynamics trajectories. This assumption limits the capability of the model to deal with chemical reactions: proton transfer reactions like those discussed, for example, in Ref.~\\citen{zhang2020deep} and in Ref.~\\citen{andrade2020free}, are allowed, but not, in general, electron transfer reactions, in which an electron is donated by one atomic entity to another. The centroid assumption shall be relaxed,\nwithin the limits of adiabatic ground-state dynamics, in a future generalization of the DPLR model.  \n}\n\n\n\n\\section*{Data availability}\n\\recheck{All data and codes needed to reproduce this work, including DP, DW, and DPLR models, are publicly available at \\url{https:\/\/doi.org\/10.5281\/zenodo.6024644}.}\n\n\\section*{Acknowledgement}\nThis work was supported by the Computational Chemical Sciences Center ``Chemistry in Solution and at Interfaces\" funded by the US Department of Energy under Award No. DE-SC0019394.  \nThe work of H.W.~was supported by the National Science Foundation of China under Grant No.11871110 and 12122103.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $E$ be an elliptic curve defined over the rational field $\\mathbb{Q}$.\nFor a prime $p$ where $E$ has good reduction, we let $E_p$\ndenote the reduced curve modulo $p$ and $\\#E_p(\\mathbb{F}_p)$\nthe number of $\\mathbb{F}_p$-rational points.\nThen  the trace of the Frobenius morphism at $p$, $a_p(E)$, satisfies\nthe well-known identity\n$\\#E_p(\\mathbb{F}_p)=p+1-a_p(E)$ and the Hasse bound\n$|a_p(E)|<2\\sqrt p$.\n\nLet $N$ be a positive integer. We are interested in the number of primes for\nwhich $\\#E_p(\\mathbb{F}_p)=N$.  In particular, we are interested in the behavior of the prime\ncounting function\n\\begin{equation*}\nM_E(N):=\\#\\{p: \\#E_p(\\mathbb{F}_p)=N\\}.\n\\end{equation*}\nNote that if $\\#E_p(\\mathbb{F}_p)=N$, then the Hasse bound implies\n$(\\sqrt p-1)^2<N<(\\sqrt p+1)^2$,\nwhich in turn implies that\n\\begin{equation*}\\label{p range}\nN^-:=(\\sqrt N-1)^2<p<(\\sqrt N+1)^2=:N^+.\n\\end{equation*}\nHence, $M_E(N)$ is a finite number, and we have the trivial bound\n\\begin{equation}\\label{trivial bound}\nM_E(N)\\ll \\frac{\\sqrt N}{\\log(N+1)}.\n\\end{equation}\nIn~\\cite{Kow:2006}, Kowalski shows that if $E$ possesses complex multiplication (CM), then\n\\begin{equation}\\label{Kowalski's question}\nM_E(N)\\ll_{E,\\varepsilon}N^\\varepsilon\n\\end{equation}\nfor any $\\varepsilon > 0$.\nHe asks if the same might be true for curves without CM.\nHowever, no bound between~\\eqref{trivial bound}\nand~\\eqref{Kowalski's question} is known for curves without CM.\n\nGiven an integer $N$, it is always possible through a Chinese Remainder Theorem\nargument to find an elliptic curve $E$ that achieves the upper bound~\\eqref{trivial bound}, i.e.,\nsuch that $\\#E_p(\\mathbb{F}_p)=N$ for every prime $p$ in the interval $(N^-,N^+)$.\nYet, for a fixed curve, one expects $M_E(N)$ to be quite small.\nConsider the following na\\\"ive probabilistic model for $M_E(N)$.\nIf we suppose that the values of $\\# E(\\mathbb{F}_p)$ are uniformly distributed, i.e., that\n\\begin{equation} \\label{naivemodel}\n\\mbox{Prob} \\left( \\# E(\\mathbb{F}_p) = N \\right) =\n\\begin{cases}\n\\frac{1}{4\\sqrt{p}} &\\text{if } N^-<p<N^+, \\\\\n 0 & \\text{otherwise,}\n\\end{cases}\n\\end{equation}\nthen we expect that\n\\begin{equation}\\label{expect M}\n\\begin{split}\nM_E(N)\n&\\approx\\sum_{p} \\mbox{Prob} \\left( \\# E(\\mathbb{F}_p) = N \\right) = \\sum_{N^-<p<N^+} \\frac{1}{4 \\sqrt{p}} \\\\\n&\\approx\\frac{1}{4 \\sqrt{N}} \\int_{N^-}^{N^+} \\frac{\\mathrm{d} t}{\\log{t}} \\sim \\frac{1}{\\log{N}}.\n\\end{split}\n\\end{equation}\nMoreover, it is quite easy to show (see~\\cite{Kow:2006} for example) that\n\\begin{equation}\\label{reg avg of M}\n\\sum_{N\\le X}M_E(N)=\\pi(X)+O(\\sqrt X)\\sim\\frac{X}{\\log X},\n\\end{equation}\nwhere as usual, $\\pi(X):=\\#\\{p\\le X: p \\text{ is prime}\\}$.\nTherefore, the average order of $M_E(N)$ is $1\/\\log N$\nin accordance with the above model.\nPerhaps the correct way to interpret these statements is to say that\n$M_E(N)$ must be equal to zero on a density one subset of the integers for the mere reason that\nthe primes are a subset of the integers of density zero.\nFinally, we note that while it is not difficult to see that $\\liminf M_E(N)=0$, \nnumerical computations~\\cite{Kow:2006} are consistent with the possibility that $\\limsup M_E(N)=\\infty$.\nIn fact, using the model~\\eqref{naivemodel} as in~\\cite{Kow:2006}, it is possible to predict this.\n\n\\section{Statement of results}\n\nIn this paper, we study the average for $M_E(N)$  \nover all elliptic curves over $\\mathbb{Q}$ (and not over $N$ as in (\\ref{reg avg of M})).\nGiven integers $a$ and $b$, let $E_{a,b}$ be the elliptic curve defined by the\nWeierstrass equation\n\\begin{equation*}\nE_{a,b}: y^2=x^3+ax+b.\n\\end{equation*}\nFor $A,B>0$, we define a set of Weierstrass equations by\n\\begin{equation*}\n\\mathscr{C}(A,B):=\\{E_{a,b}: |a|\\le A, |b|\\le B, \\Delta(E_{a,b})\\ne 0\\}.\n\\end{equation*}\nThe following is our first main result.\n\n\\begin{thm}\\label{upper bound thm}\nIf $A,B\\ge \\sqrt N\\log N$ and $AB\\ge N^{3\/2}(\\log N)^2$, then\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\\ll \\frac{\\log{\\log{N}}}{\\log{N}}\n\\end{equation*}\nholds uniformly for $N\\ge 3$.\n\\end{thm}\n\\begin{rmk}\nWe refer to the expression on the left hand side of the above inequality as the average order of\n$M_E(N)$ taken over the family $\\mathscr{C}(A,B)$.  \n\\end{rmk}\n\n\nUnder an additional hypothesis  concerning the\nshort interval distribution of primes in arithmetic progressions, we can prove an asymptotic\nformula for the average order of $M_E(N)$ over $\\mathscr{C}(A,B)$.\nIn particular, we note that all of the primes counted\nby $M_E(N)$ are of size $N$ lying in an interval of length $4\\sqrt N$.\nTherefore, we require an appropriate\nshort interval version of the Barban-Davenport-Halberstam Theorem.\n\nGiven real parameters $X,Y>0$ and integers $q$ and $a$,\nwe let $\\theta(X,Y;q,a)$ denote the\nweighted prime counting function\n\\begin{equation*}\n\\theta(X,Y;q,a):=\\sum_{\\substack{X<p\\le X+Y\\\\ p\\equiv a\\pmod q}}\\log p,\n\\end{equation*}\nand we let $E(X,Y;q,a)$ be the error in approximating $\\theta(X,Y;q,a)$ by $Y\/\\varphi(q)$.\nThat is,\n\\begin{equation*}\nE(X,Y;q,a):=\\theta(X,Y;q,a)-\\frac{Y}{\\varphi(q)}.\n\\end{equation*}\n\n\\begin{conj}\\label{bdh conj}\n(Barban-Davenport-Halberstam for intervals of length $X^\\eta$)\n\\label{shortBDH} Let $0 < \\eta \\leq 1$, and let $\\beta > 0$ be arbitrary. Suppose\nthat $X^\\eta \\leq Y \\leq X$, and that $Y \/ (\\log{X})^\\beta \\leq Q \\leq Y.$\nThen\n\\begin{align*}\n\\sum_{q \\leq Q}\\sum_{\\substack{a=1\\\\ (a,q)=1}}^{q}\\left| E(X,Y;q,a)\\right|^2 \\ll YQ \\log{X}.\n\\end{align*}\n\\end{conj}\n\\begin{rmk}\nIf $\\eta=1$, this is essentially the classical Barban-Davenport-Halberstam Theorem.\nSee for example~\\cite[p.~196]{Dav:1980}.\nThe best results known are due to Languasco, Perelli, and Zaccagnini~\\cite{LPZ:2010} who\nshow that for any $\\epsilon>0$, Conjecture~\\ref{shortBDH} holds unconditionally for\n$\\eta = 7\/12 + \\epsilon$ and for $\\eta=1\/2+\\epsilon$ under the\nGeneralized Riemann Hypothesis.\nFor our application, we essentially need $\\eta=1\/2-\\epsilon$.\n\\end{rmk}\n\n\\begin{thm}\\label{main thm}\nLet $\\gamma>0$,\nand assume that Conjecture~\\ref{bdh conj} holds with\n$\\eta=\\frac{1}{2}-(\\gamma+2)\\frac{\\log\\log N}{\\log N}$.\n Suppose further that $A,B\\ge \\sqrt N(\\log N)^{1+\\gamma}\\log\\log N$ and that\n$AB\\ge N^{3\/2}(\\log N)^{2+\\gamma}\\log\\log N$.\nThen for any odd integer $N$, we have\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\n=K(N)\\frac{N}{\\varphi(N)\\log N}+O\\left(\\frac{1}{(\\log N)^{1+\\gamma}}\\right),\n\\end{equation*}\nwhere\n\\begin{equation*}\nK(N):=\n\t\\prod_{\\ell \\nmid N}\\left(\n\t1-\\frac{\\leg{N-1}{\\ell}^2\\ell+1}{(\\ell-1)^2(\\ell+1)}\n\t\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\nmid\\nu_\\ell(N)}}\n\t\\left(1-\\frac{1}{\\ell^{\\nu_\\ell(N)}(\\ell-1)}\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\mid\\nu_\\ell(N)}}\n\t\\left(1-\\frac{\\ell-\\leg{-N_\\ell}{\\ell}}\n\t{\\ell^{\\nu_\\ell(N)+1}(\\ell-1)}\\right),\n\\end{equation*}\n$\\nu_\\ell$ denotes the usual $\\ell$-adic valuation, and\n$N_\\ell:=N\/\\ell^{\\nu_\\ell(N)}$ denotes the $\\ell$-free part of $N$.\n\\end{thm}\n\\begin{rmk}\nWe note that $K(N)$ is uniformly bounded as a function of $N$.\nWe also note that $N\/\\varphi(N)\\ll\\log\\log N$ (see~\\cite[Theorem 328]{HW:1979} for example), which\ngives the upper bound of Theorem \\ref{upper bound thm}.\nWorking with V. Chandee and D. Koukoulopoulos, the authors have recently shown that the upper bound \nimplicit in Theorem~\\ref{main thm} holds unconditionally.  That is, Theorem~\\ref{upper bound thm} holds \nwith $\\log\\log N$ replaced by $N\/\\varphi(N)$.\n\\end{rmk}\n\n\nThe average of Theorem~\\ref{main thm} displays some\ninteresting characteristics that are not present in the average order~\\eqref{reg avg of M}.\nIn particular, the main term of the average in Theorem~\\ref{main thm}\ndoes not depend solely on the size of the integer $N$ but also on some arithmetic properties of\n$N$ as it involves the factor $K(N)N\/\\varphi(N)$.\nThe occurrence of the weight $\\varphi(N)$ appearing in the denominator  seems to suggest that\nthis is another example of the Cohen-Lenstra Heuristics~\\cite{CL:1984,CL:1984-2},\nwhich predict that random groups $G$ occur with probability weighted by $1\/\\#\\mathrm{Aut}(G)$.\nNotice that if as an additive group $E(\\mathbb{F}_p) \\simeq\\mathbb{Z}\/N\\mathbb{Z}$,\nthen $\\#\\mathrm{Aut}(E(\\mathbb{F}_p)) = \\varphi(N)$.  Indeed, the Cohen-Lenstra Heuristics predict that\nrelative to other groups of same size, the cyclic groups are the most likely to occur since they have the \nfewest number of automorphisms.\n\nIn some recent work, the authors explored this connection further by\nconsidering the average of\n\\begin{equation*}\nM_E(G) := \\# \\left\\{ p : E(\\mathbb{F}_p) \\simeq G \\right\\}\n\\end{equation*}\nfor those Abelian groups $G$ which may arise as the group of $\\mathbb{F}_p$-rational points of\nan elliptic curve.  \nThis is the subject of a forthcoming paper~\\cite{DS2}.\nGiven an elliptic curve $E$, it is well-known that\n\\begin{equation*}\nE(\\mathbb{F}_p) \\simeq \\mathbb{Z}\/N_1 \\mathbb{Z} \\times \\mathbb{Z} \/ N_1 N_2 \\mathbb{Z},\n\\end{equation*}\nfor some positive integers $N_1, N_2$ satisfying the Hasse bound: $|p+1-N_1^2 N_2| \\leq 2 \\sqrt{p}$.\nUnder Conjecture~\\ref{bdh conj}, it is shown in~\\cite{DS2} that\nfor every odd order group $G = \\mathbb{Z}\/N_1 \\mathbb{Z} \\times \\mathbb{Z}\/N_1 N_2 \\mathbb{Z}$, we have that\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\\sum_{E\\in\\mathscr{C}(A,B)}M_E(G) \\sim\n\t K(G) \\frac{\\# G}{\\# \\mathrm{Aut}(G)\\log(\\#G)},\n\\end{equation*}\nprovided that $A,B$, and the exponent of $G$ (the size of the largest cyclic subgroup) are large enough with \nrespect to $\\#G=N_1^2 N_2$.  The function $K(G)$ is explicitly computed and shown to be \nnon-zero and absolutely bounded as a function of $G$.\n\n\nWe can express the results of Theorem~\\ref{main thm} as stating that for a ``random curve\"\n$E\/\\mathbb{Q}$ and a ``random prime\" $p \\in (N^-, N^+)$,\n\\begin{eqnarray*}\n\\mathrm{Prob}\\left(\\#E(\\mathbb{F}_p) = N \\right)\n\\approx \\frac{K(N) \\frac{N}{\\varphi(N) \\log(N)}}{ \\frac{4 \\sqrt{N}}{\\log{N}}}\n= \\frac{K(N) N}{\\varphi(N)} \\frac{1}{4\\sqrt{N}}\n\\end{eqnarray*}\nrefining the na\\\"ive model given by~\\eqref{naivemodel}.\nHere as in~\\eqref{naivemodel}, we make the assumption that there are about\n$4\\sqrt N\/\\log N$ primes in the interval $(N^-,N^+)$ though we can not justify such an\nassumption even under the Riemann Hypothesis.\n\nThere are many open conjectures about the distributions of invariants associated with the \nreductions of a fixed elliptic curve over the finite fields $\\mathbb{F}_p$ such as the famous conjectures \nof Koblitz~\\cite{Kob:1988} and of Lang and Trotter~\\cite{LT:1976}.\nThe Koblitz Conjecture concerns the number of primes $p\\le X$ such that $\\#E(\\mathbb{F}_p)$ is prime.\nThe fixed trace Lang-Trotter Conjecture concerns the number of primes $p\\le X$ such that\nthe trace of Frobenius $a_p(E)$ is equal to a fixed integer $t$.  Another \nconjecture of Lang and Trotter (also called the Lang-Trotter Conjecture) \nconcerns the number of primes\n$p\\le X$ such that the Frobenius field $\\mathbb{Q}(\\sqrt{a_p(E)^2-4p})$ is a fixed \nimaginary quadratic field $K$.\n\nThese conjectures are all completely open.\nTo gain evidence, it is natural to consider the averages for these conjectures over \nsome family of elliptic curves.\nThis has been done by various authors originating with the work of Fouvry and \nMurty~\\cite{FM:1996} for the number of supersingular primes (i.e., the fixed trace Lang-Trotter \nConjecture for $t=0$).  See~\\cite{DP:1999,DP:2004,Jam:2004,BBIJ:2005,JS:2011,CFJKP:2011} \nfor other averages regarding the fixed trace Lang-Trotter Conjecture.\nThe average order for the Koblitz Conjecture was considered in~\\cite{BCD:2011}.\nVery recently, the average has been successfully carried out for the Lang-Trotter Conjecture\non Frobenius fields~\\cite{CIJ:pp}.\nThe average order that we consider in this paper displays a very different character than \nthe above averages.   This is primarily because the size of primes considered varies with the \nparameter $N$. Moreover, they all must lie in a very short interval.\nThis necessitates the use of a short interval version of the Barban-Davenport-Halberstam Theorem \n(Conjecture~\\ref{bdh conj} above).\nThis is also the first time that one observes a Cohen-Lenstra phenomenon governing the \ndistribution of the average.\n\n\n\\section{Acknowledgement}\nThe authors would like to thank K. Soundararajan for pointing out how to improve\nthe average upper bound of Theorem \\ref{upper bound thm} by using short Euler products\nholding for almost all characters. They also thank Henri Cohen, Andrew Granville and\nDimitris Koukoulopoulos for useful discussions related to this work and Andrea  Smith for a\ncareful reading of the manuscript.\n\n\n\\section{Reduction to an average of class numbers}\n\nGiven a (not necessarily fundamental) discriminant $D<0$, we follow Lenstra~\\cite{Len:1987}\nin defining the \\textit{Kronecker class number} of discriminant $D$ by\n\\begin{equation}\\label{defn of K-class no}\nH(D):=\\sum_{\\substack{f^2|D\\\\ \\frac{D}{f^2}\\equiv 0,1\\pmod{4}}}\\frac{h(D\/f^2)}{w(D\/f^2)},\n\\end{equation}\nwhere $h(d)$ denotes the (ordinary) class number of the unique imaginary quadratic\norder of discriminant $d<0$ and $w(d)$ denotes the cardinality of its unit group.\n\\begin{thm}[Deuring]\\label{deuring}\nLet $p>3$ be a prime and $t$ an integer such that $t^2-4p<0$.\nThen\n\\begin{equation*}\n\\sum_{\\substack{\\tilde{E}\/\\mathbb{F}_p\\\\ a_p(E)=t}}\\frac{1}{\\#\\mathrm{Aut}(E)}\n=H(t^2-4p),\n\\end{equation*}\nwhere the sum is over the $\\mathbb{F}_p$-isomorphism classes of elliptic curves.\n\\end{thm}\n\\begin{proof}\nSee~\\cite[p.~654]{Len:1987}.\n\\end{proof}\n\nThe first step in computing the average order of $M_E(N)$ over $\\mathscr{C}(A,B)$ is to reduce to an\naverage of class numbers by using Deuring's Theorem. The following estimate\nwill then be crucial to obtain the upper bound of Theorem~\\ref{upper bound thm}, and is also used\nin getting an optimal average length in Theorem~\\ref{main thm}.\n\n\\begin{prop}\\label{upper bound for H on avg}\nFor primes $p$ in the range $N^-<p<N^+$, we define the quadratic polynomial\n\\begin{equation}\\label{defn of D}\nD_N(p):=(p+1-N)^2-4p=p^2-2(N+1)p+(N-1)^2.\n\\end{equation}\nThen\n\\begin{equation} \\label{sumofthethm}\n\\sum_{N^-<p<N^+}H\\left(D_N(p)\\right)\\ll \\frac{N \\log{\\log{N}}}{\\log{N}}.\n\\end{equation}\n\\end{prop}\n\nBefore giving the proof of this result, we define\nsome notation that we will use throughout the remainder of the article.\nGiven a negative discriminant $d$, we write $\\chi_d$ for the Kronecker symbol $\\leg{d}{\\cdot}$.\nSince $d$ is not a perfect square, the Dirichlet $L$-series defined by\n\\begin{equation*}\nL(s,\\chi_d):=\\sum_{n=1}^\\infty\\frac{\\chi_d(n)}{n^s}\n\\end{equation*}\nconverges at $s=1$.\nFinally, given a positive integer $f$, we let\n\\begin{equation}\\label{defn of d}\nd_{N,f}(p):=\\frac{D_N(p)}{f^2},\n\\end{equation}\nwhere $D_N(p)$ is defined by~\\eqref{defn of D}.\n\n\\begin{proof}[Proof of Proposition~\\ref{upper bound for H on avg}.]\nThe definition of the Kronecker class number~\\eqref{defn of K-class no} and the\nclass number formula~\\cite[p.~515]{IK:2004}\n\\begin{equation}\\label{class no formula}\n\\frac{h(d)}{w(d)}=\\frac{\\sqrt{|d|}}{2\\pi}L(1,\\chi_d)\n\\end{equation}\ngive us the identity\n\\begin{equation*}\n\\sum_{N^-<p<N^+} H(D_N(p))\n= \\sum_{N^-<p<N^+}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{\\sqrt{|D_N(p)|}}{2\\pi f}L(1,\\chi_{d_{N,f}(p)}).\n\\end{equation*}\nSince $|D_N(p)|\\le 4N$, this yields\n\\begin{equation}\\label{use bound for D}\n\\begin{split}\n\\sum_{N^-<p<N^+} H(D_N(p))\n&\\ll\\sqrt N\\sum_{N^-<p<N^+}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{L(1,\\chi_{d_{N,f}(p)})}{f}.\n\\end{split}\n\\end{equation}\n\n\nIn order to obtain an optimal bound for this expression, we will use the fact that\nfor almost all primitive Dirichlet characters $\\psi$, $L(1,\\psi)$ is well-approximated by a very\nshort Euler product.  More precisely, fix any integer $\\alpha\\ge1$.\nThen by~\\cite[Proposition 2.2]{GS:2003}, we know that\n\\begin{equation} \\label{shortEP}\nL(1, \\psi) = \\prod_{\\ell \\leq (\\log{Q})^\\alpha} \\left( 1 - \\frac{\\psi(\\ell)}{\\ell} \\right)^{-1}\n\t \\left( 1 + \\underline{o}(1) \\right)\n\\end{equation}\nfor all but at most $Q^{2\/\\alpha + 5 \\log{\\log{\\log{Q}}}\/\\log{\\log{Q}}}$ of the primitive characters of conductor less than $Q$. We remark that this gives a good upper bound for\n$L(1,\\chi)$ whenever $\\chi$ is a Dirichlet character modulo $q\\le Q$ which is induced by\na primitive character $\\psi$ satisfying~\\eqref{shortEP}.\nIndeed, let $\\chi$ be such a Dirichlet character, and let $\\psi$ be the primitive character that\ninduces $\\chi$. Then by~\\eqref{shortEP}, we have\n\\begin{equation*}\n\\begin{split}\nL(1, \\chi)\n&= \\prod_{\\ell \\mid q} \\left( 1 - \\frac{\\psi(\\ell)}{\\ell} \\right)\n \\prod_{\\ell\\le (\\log{Q})^\\alpha}\\left( 1-\\frac{\\psi(\\ell)}{\\ell}\\right)^{-1}\\left(1+\\underline{o}(1) \\right)\\\\\n&=\\prod_{\\substack{\\ell\\mid q\\\\ \\ell > (\\log{Q})^\\alpha}} \\left(1-\\frac{\\psi(\\ell)}{\\ell} \\right)\n\\prod_{\\ell \\leq (\\log{Q})^\\alpha} \\left( 1 - \\frac{\\chi(\\ell)}{\\ell} \\right)^{-1}\n\\left(1+\\underline{o}(1) \\right)\\\\\n&\\ll\\prod_{\\substack{\\ell\\mid q\\\\ \\ell > (\\log{Q})^\\alpha}} \\left(1+\\frac{1}{\\ell} \\right)\n\\prod_{\\ell \\leq (\\log{Q})^\\alpha} \\left( 1 - \\frac{1}{\\ell} \\right)^{-1}\\\\\n&\\ll\\prod_{\\substack{\\ell\\mid q\\\\ \\ell > (\\log{Q})^\\alpha}} \\left(1+\\frac{1}{\\ell} \\right)\\log\\log Q,\n\\end{split}\n\\end{equation*}\nwhere the last line follows by Mertens' formula~\\cite[p.~34]{IK:2004} since $\\alpha$ is fixed.\nFor the remaining product, we observe that\n\\begin{equation*}\n\\prod_{\\substack{\\ell\\mid q\\\\ \\ell > (\\log{Q})^\\alpha}} \\left(1+\\frac{1}{\\ell} \\right)\n\\le\\exp\\left\\{\\sum_{\\substack{\\ell\\mid q\\\\ \\ell>(\\log Q)^\\alpha}}\\frac{1}{\\ell}\\right\\}\n\\le\\exp\\left\\{\\frac{\\omega(q)}{(\\log Q)^{\\alpha}}\\right\\}\n\\le\\exp\\left\\{\\frac{(\\log q)^{1-\\alpha}}{\\log 2}\\right\\},\n\\end{equation*}\nwhere $\\omega(q)$ denotes the number of distinct prime factors of $q$.\nTherefore, since $\\alpha\\ge1$,  we may conclude that\nif $\\chi$ is a character of modulus $q\\le Q$ and~\\eqref{shortEP} holds for the primitive\ncharacter inducing $\\chi$, then\n\\begin{equation}\\label{nexcept-L bound}\nL(1,\\chi)\\ll\\log\\log Q.\n\\end{equation}\n\nWe make use of this fact in~\\eqref{use bound for D} as follows.\nLet $d_N^*(p)$ be the discriminant of the imaginary\nquadratic field $\\mathbb{Q}(\\sqrt{D_N(p)})$.  Then $d_N^*(p)$ is a fundamental discriminant,\nand $\\chi_{d_N^*(p)}$ is the primitive character inducing every character of the set\n$\\{\\chi_{d_{N,f}(p)}: f^2\\mid D_N(p)\\}$.  Furthermore, $|d_N^*(p)|$ is the conductor of\neach of these characters, and $3\\le |d_N^*(p)|\\le 4N$.\nNow fix some $\\alpha>100$, and\nlet $\\mathscr E(Q)$ be the set of primitive characters of conductor less than or equal to $Q$\nfor which~\\eqref{shortEP} does not hold.  Then $\\# \\mathscr E(4N)\\ll N^{1\/50}$.\nWe now divide the outer sum over $p$ on the right-hand side of ~\\eqref{use bound for D} \naccording to whether or not the primitive character $\\chi_{d_N^*(p)}$ is in the\nexceptional set $\\mathscr E(4N)$.  For those $p$ for which $\\chi_{d_N^*(p)}$ is not\nexceptional, we use~\\eqref{nexcept-L bound}, writing\n\\begin{equation*}\n\\sum_{\\substack{N^-<p<N^+\\\\ \\chi_{d_N^*(p)} \\not\\in \\mathscr{E}(4N)}}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{L(1,\\chi_{d_{N,f}(p)})}{f}\n\\ll\\log{\\log{N}}\\sum_{f\\le 2\\sqrt N}\\frac{1}{f}\\sum_{\\substack{N^-<p<N^+\\\\  {f^2\\mid D_N(p)}}} 1.\n\\end{equation*}\nTo bound the sum over $p$, we apply the Brun-Titchmarsh inequality~\\cite[p.~167]{IK:2004},\nwhich gives that\n\\begin{equation*}\n\\#\\{N^-<p<N^+ : p\\equiv a\\pmod f\\}\n\\ll\\frac{\\sqrt N}{\\varphi(f)\\log(4\\sqrt N\/f)}.\n\\end{equation*}\nFor the sum over $a$, we use the bound\n\\begin{equation*}\n\\#\\{a\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(a)\\equiv 0\\pmod f\\}\\ll\\sqrt f,\n\\end{equation*}\nwhich is Lemma~\\ref{bound quad cong soln count} of\nSection~\\ref{proofs of lemmas}.\nCombining these two estimates, we have that\n\\begin{equation}\\label{nexcept-sum bound}\n\\begin{split}\n\\sum_{\\substack{N^-<p<N^+\\\\ \\chi_{d_N^*(p)} \\not\\in \\mathscr{E}(4N)}}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{L(1,\\chi_{d_{N,f}(p)})}{f}\n&\\ll \\frac{\\sqrt N\\log{\\log{N}}}{\\log{N}} \\sum_{f\\ge 1}\\frac{\\log f}{f^{1\/2}\\varphi(f)}\\\\\n&\\ll\\frac{\\sqrt N\\log{\\log{N}}}{\\log{N}}.\n\\end{split}\n\\end{equation}\n\nIt remains to estimate the sum over primes $p$ such that\n$\\chi_{d_N^*(p)} \\in \\mathscr{E}(4N)$.\nIn that case, we simply need the standard bound\n\\begin{equation*}\nL(1, \\chi) \\ll \\log{q},\n\\end{equation*}\nwhich is valid for all Dirichlet characters of conductor $q$ \n(see~\\cite[p.~96]{Dav:1980} for example).\nWe note that\n\\begin{equation*}\n\\#\\{N^-<p<N^+: \\chi_{d_N^*(p)}\\in\\mathscr E(4N)\\}\\le\\# \\mathscr E(4N)\\tau(4N),\n\\end{equation*}\nwhere $\\tau(n)$ denotes the number of positive divisors of $n$.\nTherefore, we obtain the bound\n\\begin{equation}\\label{except-sum bound}\n\\begin{split}\n\\sum_{\\substack{N^-<p<N^+\\\\ \\chi_{d_N^*(p)}\\in \\mathscr{E}(4N)}}\n\\sum_{\\substack{f^2\\mid D_N(p)\\\\ \\frac{D_N(p)}{f^2}\\equiv 0,1\\pmod 4}}\n\t\\frac{L(1,\\chi_{d_{N,f}(p)})}{f}\n&\\ll\\log{N} \\sum_{\\substack{N^-<p<N^+\\\\ \\chi_{d_N^*(p)}\\in\\mathscr{E}(4N)}}\n\t\\sum_{f^2 \\mid D_N(p)} \\frac{1}{f}\\\\\n&\\ll N^{1\/50+\\varepsilon} \n\\end{split}\n\\end{equation}\nfor any $\\varepsilon > 0$.\nCombining~\\eqref{use bound for D},~\\eqref{nexcept-sum bound}, and~\\eqref{except-sum bound}\ncompletes the proof of Proposition~\\ref{upper bound for H on avg}.\n\\end{proof}\n\n\\begin{prop}\\label{average order in terms of class numbers}\nLet $D_N(p)$ be as defined by~\\eqref{defn of D}. Then\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\n=\\sum_{N^-<p<N^+}\\frac{H(D_N(p))}{p}+\\mathcal E(N;A,B),\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\mathcal E(N;A,B)\\ll\n\\frac{\\log\\log N}{N\\log N}+\\left(\\frac{1}{A}+\\frac{1}{B}\\right)\\sqrt N\\log\\log N+\\frac{N^{3\/2}\\log N\\log\\log N}{AB}\n\\end{equation*}\nuniformly for $A,B>0$.\n\\end{prop}\n\\begin{rmk}\nThe above holds without assuming that $N$ is odd.\n\\end{rmk}\n\n\\begin{proof}[Proof of Proposition~\\ref{average order in terms of class numbers}]\nFirst, we write $M_E(N)$ as a sum over primes and interchange sums\nto obtain\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\n\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\n&=\n\\frac{1}{\\#\\mathscr{C}(A,B)}\n\\sum_{N^-<p<N^+}\n\\sum_{\\substack{E\\in\\mathscr{C}(A,B)\\\\ \\#E_p(\\mathbb{F}_p)=N}}1.\\\\\n\\end{split}\n\\end{equation*}\nFor each prime $p$, we group the $E\\in\\mathscr{C}(A,B)$ according to which isomorphism\nclass they reduce modulo $p$, writing\n\\begin{equation*}\n\\sum_{\\substack{E\\in\\mathscr{C}(A,B)\\\\ \\#E_p(\\mathbb{F}_p)=N}}1\n=\\sum_{\\substack{\\tilde E\/\\mathbb{F}_p\\\\ \\#\\tilde E(\\mathbb{F}_p)=N}}\n\t\\#\\{E\\in\\mathscr{C}(A,B): E_p\\cong\\tilde E\\}.\n\\end{equation*}\nFor $N$ large enough ($N\\ge 8$), the primes $2$ and $3$ will not enter into\nthe sum over $p$.  Thus, we will assume\nthat $p>3$ throughout the remainder of the article.\nTherefore, given a elliptic curve defined over $\\mathbb{F}_p$, we may associate\na Weierstrass equation, say $E_{s,t}: y^2=x^3+sx+t$ with $s,t\\in\\mathbb{F}_p$.\nUsing a character sum argument as in~\\cite[pp.~93-95]{FM:1996}, we have\n\\begin{equation*}\n\\begin{split}\n\\#\\{E\\in\\mathscr{C}(A,B): E_p\\cong E_{s,t}\\}\n=\\frac{4AB}{\\#\\mathrm{Aut}(E_{s,t})p}\n&+O\\left(\\frac{AB}{p^2}+\\sqrt p(\\log p)^2\\right)\\\\\n&+\\begin{cases}\nO\\left(\\frac{A\\log p}{\\sqrt p}+\\frac{B\\log p}{\\sqrt p}\\right)&\\text{if }st\\ne 0,\\\\\nO\\left(\\frac{A\\log p}{\\sqrt p}+B\\log p\\right)&\\text{if }s=0,\\\\\nO\\left(A\\log p+\\frac{B\\log p}{\\sqrt p}\\right)&\\text{if }t=0.\n\\end{cases}\n\\end{split}\n\\end{equation*}\nHere $\\mathrm{Aut}(E_{s,t})$ denotes the size of the automorphism group of $E_{s,t}$ over $\\mathbb{F}_p$.\nSubstituting this estimate and applying Theorem~\\ref{deuring}, we find that\n\\begin{equation*}\n\\frac{1}{\\#\\mathscr{C}(A,B)}\n\\sum_{E\\in\\mathscr{C}(A,B)}M_{E}(N)\n=\n\\sum_{N^-<p<N^+}\\frac{H(D_N(p))}{p}+\\mathcal E(N;A,B),\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\begin{split}\n\\mathcal E(N;A,B)\n&\\ll\\left\\{\\frac{1}{N^2}\n\t+\\frac{\\log N}{\\sqrt N}\\left(\\frac{1}{A}+\\frac{1}{B}\\right)\n\t+\\frac{\\sqrt N(\\log N)^2}{AB}\\right\\}\n\t\\sum_{N^-<p<N^+}H(D_N(p))\\\\\n&+\\left(\\frac{1}{A}+\\frac{1}{B}\\right)\\log N\\sum_{N^-<p<N^+}1.\n\\end{split}\n\\end{equation*}\nIn estimating the error, we have used the facts that $\\#\\mathscr{C}(A,B)=4AB+O(A+B)$, $p=N+O(\\sqrt N)$ for every\nprime $p$ in the interval $(N^-,N^+)$, and there are at most $10$ isomorphism classes $E_{s,t}$ over \n$\\mathbb{F}_p$ with $st=0$.\nThe result now follows by applying Proposition~\\ref{upper bound for H on avg} and the \nupper bound $\\{N^-<p<N^+\\}\\ll\\sqrt N\/\\log N$.\n\\end{proof}\n\n\nTheorem~\\ref{upper bound thm} now follows immediately upon combining\nProposition ~\\ref{upper bound for H on avg} and\nProposition~\\ref{average order in terms of class numbers}, and noting again that\n$p=N+O(\\sqrt N)$ for every prime $p$ in the interval\n$(N^-,N^+)$.\nFurthermore, we have reduced the proof of Theorem~\\ref{main thm} to computing\nthe sum\n\\begin{equation*}\n\\sum_{N^-<p<N^+}\\frac{H(D_N(p))}{p}.\n\\end{equation*}\nThis computation requires several intermediate results.\nTherefore, we delay it until Section~\\ref{proof of main thm}.\n\n\n\\section{A short average of special values of Dirichlet $L$-functions}\n\nSince Theorem~\\ref{main thm} holds only for odd $N$, we assume for the\nthe remainder of the article that $N$ is an odd integer except during\nthe proof of Lemma~\\ref{bound quad cong soln count}.  Recall that\nLemma~\\ref{bound quad cong soln count} was used in the proof of\nProposition~\\ref{upper bound for H on avg}, which holds for all positive integers $N$.\n\nIn this section, we prove a short average result for special values of Dirichlet\n$L$-functions that is needed to compute our average over elliptic curves. As the average\nis very short, we need that the equivalent of the Barban-Davenport-Halberstam Theorem holds\nfor intervals of that size.  \n\n\n\\begin{thm} \\label{thmaverageSV}\nLet $\\gamma>0$.\nSuppose that $N^-\\le X<X+Y\\le N^+$ with $Y\\gg \\sqrt N\/(\\log N)^\\upsilon$ for some choice of \n$\\upsilon\\ge 0$.\nAssume that Conjecture~\\ref{shortBDH} holds for intervals of length $Y$.\nThen for odd integers $N$,\n\\begin{equation*}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p = K_0(N)Y + O \\left( \\frac{Y}{(\\log{N})^\\gamma} \\right),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{K(N) as a double infinite sum}\nK_0(N):=\n\\sum_{\\substack{f=1\\\\ (f,2)=1}}^\\infty\\frac{1}{f}\\sum_{n=1}^\\infty\\frac{1}{n\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\t\\#C_N(a,n,f),\n\\end{equation}\nand\n\\begin{equation}\\label{defn of C}\nC_N(a,n,f):=\\left\\{z\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*: D_N(z)\\equiv af^2\\pmod{4nf^2}\\right\\}.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nLet $U$ be a real parameter to be determined.\nUsing partial summation and Burgess' bound for character sums~\\cite[Theorem 2]{Bur:1963} to\nbound the tail of the $L$-series, we have\n\\begin{equation*}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\n=\\sum_{n\\ge 1}\\leg{d_{N,f}(p)}{n}\\frac{1}{n}\n=\\sum_{n\\le U}\\leg{d_{N,f}(p)}{n}\\frac{1}{n}\n\t+O\\left(\\frac{|d_{N,f}(p)|^{\\frac{7}{32}}}{\\sqrt U}\\right).\n\\end{equation*}\nFor $N^-<p<N^+$, we have $|d_{N,f}(p)|\\le 4N\/f^2$, and hence\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\\log p\n\\sum_{n> U}\\frac{1}{n}\n\\leg{d_{N,f}(p)}{n}\n\\ll\\frac{YN^{7\/32}}{\\sqrt U}.\n\\end{split}\n\\end{equation*}\nNow let $V$ be a real parameter to be determined.\nUsing Lemma~\\ref{bound quad cong soln count}, we obtain\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{V<f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{n\\le U}&\n\\frac{1}{n}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\n\\leg{d_{N,f}(p)}{n}\\log p\\\\\n&\\ll\\log U\\log N\\sum_{\\substack{V<f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\n\t\\frac{1}{f}\\sum_{\\substack{X<k\\le X+4\\sqrt N\\\\ f|D_N(p)}}1\\\\\n&\\ll\\log U\\log N\\sum_{\\substack{V<f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\n\\frac{\\sqrt N\\#\\{z\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(z)\\equiv 0\\pmod f\\}}{f^2}\\\\\n&\\ll\\sqrt N\\log U\\log N\\sum_{f>V}\\frac{1}{f^{3\/2}}\\\\\n&\\ll\\frac{\\sqrt N\\log U\\log N}{\\sqrt V},\n\\end{split}\n\\end{equation*}\nand therefore,\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p\n&=\\sum_{\\substack{f\\le V\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{n\\le U}\n\\frac{1}{n}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\n\\leg{d_{N,f}(p)}{n}\\log p\\\\\n&\\quad+O\\left(\\frac{YN^{7\/32}}{\\sqrt U}+\\frac{\\sqrt N\\log U\\log N}{\\sqrt V}\\right).\n\\end{split}\n\\end{equation*}\nWith $C_N(a,n,f)$ as defined by~\\eqref{defn of C},\nwe regroup terms on the right hand side above, writing\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\n\\leg{d_{N,f}(p)}{n}\\log p\n&=\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{b\\in C_N(a,n,f)}\\theta(X,Y;4nf^2,b)\\\\\n&\\quad+O\\left(\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{\\substack{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}\\\\ D_N(p)\\equiv af^2\\\\ p\\mid 4nf^2}}\\log p\\right).\n\\end{split}\n\\end{equation*}\nIf $p$ satisfies the congruence\n$D_N(p)\\equiv af^2\\pmod{4nf^2}$ and $p$ divides $4nf^2$, then $p$ divides\n$(4nf^2,(N-1)^2-af^2)$.  It follows that such a $p$ must divide $n(N-1)$.\nHence, the error term\n\\begin{equation*}\n\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{\\substack{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}\\\\ D_N(p)\\equiv af^2\\\\ p\\mid 4nf^2}}\\log p\n\\ll U\\log N\\log V + U\\log U\\log V,\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p\n&=\\sum_{\\substack{f\\le V, n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{b\\in C_N(a,n,f)}\\theta(X,Y;4nf^2,b)\\\\\n&\\quad+O\\left(\n\t\t\\frac{YN^{7\/32}}{\\sqrt U}+\\frac{\\sqrt N\\log U\\log N}{\\sqrt V}\n\t\t+U\\log(UN)\\log V\\right).\n\\end{split}\n\\end{equation*}\nWe approximate $\\theta(X,Y;4nf^2,b)$ by $Y\/\\varphi(4nf^2)$, incurring an error of\n\\begin{equation}\\label{error from approximating sum over primes}\n\\sum_{\\substack{f\\le V,n\\le U\\\\ (f,2)=1}}\\frac{1}{fn}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{b\\in C_N(a,n,f)}E(X,Y; 4nf^2,b).\n\\end{equation}\nFor any given value of $b\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*$, there is at most one value of\n$a\\in\\mathbb{Z}\/4n\\mathbb{Z}$ satisfying the congruence $D_N(b)\\equiv af^2\\pmod{4nf^2}$.  Hence, interchanging the two inner sums shows that\n\\begin{equation*}\n\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\\sum_{b\\in C_N(a,n,f)}E(X,Y; 4nf^2,b)\n\\ll \\sum_{b\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*}\\left|E(X,Y;4nf^2,b)\\right|.\n\\end{equation*}\nBy Cauchy-Schwarz,\nthe error term~\\eqref{error from approximating sum over primes} is\n\\begin{equation*}\n\\begin{split}\n&\\ll\n\\sum_{f\\le V}\\frac{1}{f}\n\\sum_{n\\le U}\n\\frac{1}{n}\n\\sum_{b\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*}\\left|E(X,Y;4nf^2,b)\\right|\\\\\n&\\le\n\\sum_{f\\le V}\\frac{1}{f}\n\\left[\n\\sum_{n\\le U}\\frac{\\varphi(4nf^2)}{n^2}\n\\right]^{1\/2}\n\\left[\n\\sum_{n\\le U}\n\\sum_{b\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*}\n\\left|E(X,Y;4nf^2,b)\\right|^2\n\\right]^{1\/2}\\\\\n&\\ll\nV\\sqrt{\\log U}\n\\left[\n\\sum_{q\\le 4UV^2}\\sum_{\\substack{a=1\\\\ (a,q)=1}}^q\n\\left|E(X,Y;q,a)\\right|^2\\right]^{1\/2}.\n\\end{split}\n\\end{equation*}\nAssuming Conjecture~\\ref{shortBDH} for an appropriate value of $\\eta$, we obtain the bound\n\\begin{equation*}\nV\\sqrt{\\log U}\n\\left[\n\\sum_{q\\le 4UV^2}\\sum_{\\substack{a=1\\\\ (a,q)=1}}^q\n\\left|E(X,Y;q,a)\\right|^2\\right]^{1\/2}\n\\ll\\frac{YV\\sqrt{\\log U\\log N}}{(\\log N)^{2\\upsilon+3\\gamma+5}}\n\\end{equation*}\nwhenever\n\\begin{equation}\\label{bdh restriction on U and V}\nUV^2\\le \\frac{Y}{(\\log N)^{4\\upsilon+6\\gamma+10}}.\n\\end{equation}\nThus, we have\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p\n&=Y\\sum_{\\substack{f\\le V,n\\le U\\\\ (f,2)=1}}\\frac{1}{fn\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\t\\#C_N(a,n,f) \\\\\n&\\quad+O \\left( \\frac{YN^{7\/32}}{\\sqrt U}+\\frac{\\sqrt N\\log U\\log N}{\\sqrt V}+U\\log(UN)\\log V\\right)\\\\\n&\\quad+O\\left(\\frac{YV\\sqrt{\\log U\\log N}}{(\\log N)^{2\\upsilon+3\\gamma+5}}\\right).\n\\end{split}\n\\end{equation*}\n\n\\begin{lma}\\label{complete K sum}\nFor any $U,V,\\epsilon>0$, we have\n\\begin{equation*}\n\\begin{split}\nK_0(N)=\n\\sum_{\\substack{f\\le V,n\\le U\\\\ (f,2)=1}}\\frac{1}{fn\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\\#C_N(a,n,f)\t\n\t+O\\left(\\frac{N^\\epsilon}{\\sqrt U}+\\frac{\\log\\log N}{V}\\right).\n\\end{split}\n\\end{equation*}\n\\end{lma}\nWe delay the proof of Lemma~\\ref{complete K sum} until Section~\\ref{proofs of lemmas}.\nApplying the lemma and choosing\n\\begin{equation*}\nU=\\frac{Y}{(\\log N)^{4\\upsilon+10\\gamma+18}},\\quad V=(\\log N)^{2\\upsilon+2\\gamma +4},\n\\end{equation*}\nwe have\n\\begin{equation*}\n\\begin{split}\n\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\nL\\left(1,\\chi_{d_{N,f}(p)}\\right)\\log p\n=K_0(N)Y&\n+O\\left(\\frac{Y}{(\\log N)^\\gamma}\\right)\n\\end{split}\n\\end{equation*}\nprovided that $Y\\gg\\sqrt N\/(\\log N)^{\\upsilon}$.\nNote that our choice of $U,V$ satisfies the condition~\\eqref{bdh restriction on U and V}.\n\\end{proof}\n\n\\section{Computing the ``almost constant\"}\n\nRecall that $C_N(a,n,f)$ was defined by\n\\begin{equation*}\nC_N(a,n,f)=\\left\\{z\\in(\\mathbb{Z}\/4nf^2\\mathbb{Z})^*: D_N(z)\\equiv af^2\\pmod{4nf^2}\\right\\},\n\\end{equation*}\nwhere $D_N(z)=z^2-2(N+1)z+(N-1)^2$.\nThe following is the main result of this section.\n\\begin{prop}\\label{product formula}\nWith $K(N)$ as defined in Theorem~\\ref{main thm} and $K_0(N)$ as defined in\nTheorem~\\ref{thmaverageSV}, we have\n\\begin{equation*}\n\\frac{N}{\\varphi(N)}K(N)=K_0(N).\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nBy the Chinese Remainder Theorem and the definition of $C_N(a,n,f)$,\n\\begin{equation*}\n\\#C_N(a,n,f)=\n\\prod_{\\ell\\mid 4nf^2}\\#C_N^{(\\ell)}(a,n,f),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{crt pieces of C}\nC_N^{(\\ell)}(a,n,f):=\n\\{z\\in(\\mathbb{Z}\/\\ell^{\\nu_\\ell(4nf^2)}\\mathbb{Z})^*:\n\tD_N(z)\\equiv af^2\\pmod{\\ell^{\\nu_\\ell(4nf^2)}}\\}.\n\\end{equation}\nWe require the following lemma whose proof we delay until Section~\\ref{proofs of lemmas}.\n\n\\begin{lma}\\label{counting solutions}\nSuppose that $N$ and $f$ are odd and that $a\\equiv 1\\pmod 4$.\nLet $\\ell$ be any odd prime dividing $nf$, and let\n$e=\\nu_\\ell(4nf^2)=\\nu_\\ell(nf^2)$.\nIf $\\ell\\nmid 4N+af^2$, then\n\\begin{equation*}\n\\#C_N^{(\\ell)}(a,n,f)=\n\\begin{cases}\n1+\\leg{4N+af^2}{\\ell}&\\text{if }\\ell\\nmid (N-1)^2-af^2,\\\\\n1&\\text{if }\\ell\\mid (N-1)^2-af^2.\n\\end{cases}\n\\end{equation*}\nIf $\\ell\\mid 4N+af^2$, then with $s=\\nu_\\ell(4N+af^2)$, we have\n\\begin{equation*}\n\\begin{split}\n\\#C_N^{(\\ell)}(a,n,f)\n&=\\begin{cases}\n2\\leg{N+1}{\\ell}^2\\ell^{s\/2}\n\t&\\text{if }1\\le s<e,\\ 2\\mid s, \\text{ and }\\leg{(4N+af^2)\/\\ell^s}{\\ell}=1,\\\\\n\\leg{N+1}{\\ell}^2\\ell^{\\floor{e\/2}}\n\t&\\text{if }s\\ge e,\\\\\n0&\\text{otherwise}.\n\\end{cases}\n\\end{split}\n\\end{equation*}\nIn particular, if $\\ell\\mid f$, then\n\\begin{equation*}\n\\#C_N^{(\\ell)}(1,1,f)=\\begin{cases}\n1+\\leg{N(N-1)^2}{\\ell}&\\text{if }\\ell\\nmid N,\\\\\n2\\ell^{\\nu_\\ell(N)\/2}&\\text{if }1\\le\\nu_\\ell(N)<2\\nu_\\ell(f),\\ 2\\mid\\nu_\\ell(N),\n\t\\text{ and }\\leg{N_\\ell}{\\ell}=1,\\\\\n\\ell^{\\nu_\\ell(f)}&\\text{if }2\\nu_\\ell(f)\\le\\nu_\\ell(N),\\\\\n0&\\text{otherwise},\n\\end{cases}\n\\end{equation*}\nwhere $N_\\ell=N\/\\ell^{\\nu_\\ell(N)}$ is the $\\ell$-free part of $N$.\nFurthermore,\n\\begin{equation*}\n\\#C_N^{(2)}(a,n,f)\n=\\begin{cases}\n2&\\text{if }\\nu_2(4nf^2)=2+\\nu_2(n)=2,\\\\\n4&\\text{if }\\nu_2(4nf^2)=2+\\nu_2(n)\\ge 3\\text{ and }a\\equiv 5\\pmod 8,\\\\\n0&\\text{otherwise}.\n\\end{cases}\n\\end{equation*}\n\\end{lma}\n\nBy Lemma~\\ref{counting solutions},\nwe may write\n\\begin{equation*}\n\\#C_N^{(2)}(a,n,f)\n=2\\mathcal S_2(n,a),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{defn of S_2}\n\\mathcal S_2(n,a):=\\begin{cases}\n1&\\text{if } 2\\nmid n,\\\\\n2&\\text{if }2\\mid n\\text{ and } a\\equiv 5\\pmod 8,\\\\\n0&\\text{otherwise}.\n\\end{cases}\n\\end{equation}\n\nNote that if $\\ell$ is a prime dividing $f$ and not dividing $n$, then\n$af^2\\equiv 0\\pmod{\\ell^{\\nu_\\ell(4nf^2)}}$ as $\\nu_\\ell(4nf^2)=\\nu_\\ell(f^2)$.\nHence, in this case, $\\#C_N^{(\\ell)}(a,n,f)$ does not depend on the value of $a$, and so we write\n$\\#C_N^{(\\ell)}(a,n,f)=\\#C_N^{(\\ell)}(1,1,f)$ if $\\ell\\mid f$ and $\\ell\\nmid n$.\nTherefore, letting $n'$ denote the odd part of $n$ and\n\\begin{equation*}\nc_{N,f}(n):=\\sum_{\\substack{a\\in(\\mathbb{Z}\/4n\\mathbb{Z})^*\\\\ a\\equiv 1\\pmod 4}}\n\\leg{a}{n}\n\\mathcal S_2(n,a)\n\\prod_{\\ell\\mid n'}\n\\#C_N^{(\\ell)}(a,n,f),\n\\end{equation*}\nwe may write\n\\begin{equation}\\label{separate n and f}\n\\begin{split}\nK_0(N)\n&=\\sum_{\\substack{f=1\\\\ (f,2)=1}}^\\infty\\frac{1}{f}\\sum_{n=1}^\\infty\\frac{1}{n\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\n\t\\#C_N(a,n,f)\\\\\n&=\\sum_{\\substack{f=1\\\\ (f,2)=1}}^\\infty\\frac{1}{f^2\\varphi(f)}\n\t\\sum_{n=1}^\\infty\\frac{2\\varphi((n,f))}{(n,f)n\\varphi(4n)}\n\t\\left[\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\nmid n}}\\#C_N^{(\\ell)}(1,1,f)\\right]c_{N,f}(n)\\\\\n&=\\sum_{f=1}^\\infty\\strut^\\prime\n\t\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n=1}^\\infty\\frac{2\\varphi((n,f))}{(n,f)n\\varphi(4n)}\n\t\\left[\\prod_{\\ell\\mid (f,n)}\\#C_N^{(\\ell)}(1,1,f)\\right]^{-1}c_{N,f}(n),\n\\end{split}\n\\end{equation}\nwhere the prime on the outer sum indicates that the sum is to be restricted to those $f$ which are\nodd and are not divisible by any prime for which $\\#C_N^{(\\ell)}(1,1,f)=0$.\n\n\nIn order to proceed further, we must show how to compute the function $c_{N,f}(n)$.\nWe summarize the computation in the following lemma whose proof we also delay\nuntil Section~\\ref{proofs of lemmas}.\n\n\\begin{lma}\\label{compute factors of c}\nSuppose that $N$ and $f$ are odd.\nThe function $c_{N,f}(n)$ is multiplicative in $n$.\nLet $\\alpha$ be a positive integer and $\\ell$ an odd prime.\nThen\n\\begin{equation*}\n\\frac{c_{N,f}(2^\\alpha)}{2^{\\alpha-1}}=(-1)^\\alpha 2.\n\\end{equation*}\nIf $\\ell\\mid f$ and $\\ell\\nmid N$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\#C_N^{(\\ell)}(1,1,f)\n\\begin{cases}\n\\ell-1&\\text{if }2\\mid\\alpha,\\\\\n0&\\text{if }2\\nmid\\alpha.\n\\end{cases}\n\\end{equation*}\nIf $\\ell\\mid N$ and $\\ell\\nmid f$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}=\\ell-2.\n\\end{equation*}\nIf $\\ell\\nmid Nf$, then\n\\begin{equation*}\n\\begin{split}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n&=\\begin{cases}\n\\ell-2-\\leg{N}{\\ell}-\\leg{N^2-1}{\\ell}^2\n+\\leg{N+1}{\\ell}^2\n\t&\\text{if }2\\mid\\alpha,\\\\\n-1-\\leg{-N}{\\ell}-\\leg{N^2-1}{\\ell}^2\n+\\leg{-N(N+1)^2}{\\ell}\n\t&\\text{if }2\\nmid\\alpha\n\\end{cases}\\\\\n&=\\begin{cases}\n\\ell-1-\\leg{N}{\\ell}-\\leg{N-1}{\\ell}^2\n\t&\\text{if }2\\mid\\alpha,\\\\\n-1-\\leg{N-1}{\\ell}^2\n\t&\\text{if }2\\nmid\\alpha.\n\\end{cases}\n\\end{split}\n\\end{equation*}\nIf $\\ell\\mid (f,N)$ and $2\\nu_\\ell(f)<\\nu_\\ell(N)$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\#C_N^{(\\ell)}(1,1,f)(\\ell-1).\n\\end{equation*}\nIf $\\ell\\mid (f,N)$ and $\\nu_\\ell(N)<2\\nu_\\ell(f)$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\#C_N^{(\\ell)}(1,1,f)\\begin{cases}\n\\ell-1&\\text{if }2\\mid \\alpha,\\\\\n0&\\text{if }2\\nmid\\alpha.\n\\end{cases}\n\\end{equation*}\nIf $\\ell\\mid (f,N)$ and $\\nu_\\ell(N)=2\\nu_\\ell(f)$, then\n\\begin{equation*}\n\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\#C_N^{(\\ell)}(1,1,f)\\begin{cases}\n\\left(\\ell-1-\\leg{N_\\ell}{\\ell}+\\leg{-N_\\ell}{\\ell}\\right)&\\text{if }2\\mid\\alpha,\\\\\n\\left(\\leg{-N_\\ell}{\\ell}-1\\right)&\\text{if }2\\nmid\\alpha,\\\\\n\\end{cases}\n\\end{equation*}\nwhere $N_\\ell=N\/\\ell^{\\nu_\\ell(N)}$ denotes the $\\ell$-free part of $N$.\nFurthermore, for any $n$, we have the bound\n\\begin{equation*}\nc_{N,f}(n)\\ll\\frac{n\\prod_{\\ell\\mid (f,n)}\\#C_N(1,1,f)}{\\kappa_{2N}(n)},\n\\end{equation*}\nwhere for any integer $m$, $\\kappa_m(n)$ is the multiplicative function defined on prime\npowers by\n\\begin{equation}\\label{defn of kappa}\n\\kappa_m(\\ell^\\alpha)\n:=\\begin{cases}\n\\ell&\\text{if }2\\nmid\\alpha\\text{ and }\\ell\\nmid m,\\\\\n1&\\text{otherwise}.\n\\end{cases}\n\\end{equation}\n\\end{lma}\n\nRecalling the restrictions on $f$ in~\\eqref{separate n and f} and applying\nLemma~\\ref{compute factors of c}, the sum over $n$ in~\\eqref{separate n and f} may be factored as\n\\begin{equation*}\n\\begin{split}\n&\\sum_{n=1}^\\infty\\frac{2\\varphi((n,f))}{(n,f)n\\varphi(4n)}\n\t\\left[\\prod_{\\ell\\mid (f,n)}\\#C_N^{(\\ell)}(1,1,f)\\right]^{-1}c_{N,f}(n)\\\\\n&\\quad=\\left\\{\\sum_{\\alpha\\ge 0}\\frac{2c_{N,f}(2^\\alpha)}{2^\\alpha\\varphi(2^{\\alpha+2})}\\right\\}\n\t\\prod_{\\substack{\\ell\\nmid f\\\\ \\ell\\ne 2}}\n\t\t\\left\\{\\sum_{\\alpha\\ge 0}\\frac{c_{N,f}(\\ell^\\alpha)}{\\ell^\\alpha\\varphi(\\ell^\\alpha)}\\right\\}\n\t\\prod_{\\ell\\mid f}\n\t\t \\left\\{1+ \\sum_{\\alpha\\ge 1}\\frac{\\varphi((\\ell^\\alpha,f))c_{N,f}(\\ell^\\alpha)}\n \t\t {(\\ell^\\alpha,f)\\ell^\\alpha\\varphi(\\ell^\\alpha)\\#C_N^{(\\ell)}(1,1,f)}\\right\\}\n\\\\\n &\\quad\n =\\frac{2}{3}\n\t\\prod_{\\substack{\\ell\\nmid f\\\\ \\ell\\mid N}}F_0(\\ell)\t\n\t\\prod_{\\substack{\\ell\\nmid f\\\\ \\ell\\nmid N\\\\ \\ell\\ne 2}}F_1(\\ell)\n\t\\prod_{\\ell\\mid f}F_2(\\ell,f),\n\\end{split}\n\\end{equation*}\nwhere for any odd prime $\\ell$, we make the definitions\n\\begin{align*}\nF_0(\\ell)&:=\\left(1+\\frac{\\ell-2}{(\\ell-1)^2}\\right),\\\\\nF_1(\\ell)\n&:=\\left(1-\n\t\\frac{\\leg{N-1}{\\ell}^2\\ell\n\t+\\leg{N}{\\ell}+\\leg{N-1}{\\ell}^2+1}{(\\ell-1)(\\ell^2-1)}\n\\right),\\\\\nF_2(\\ell,f)&:=\\begin{cases}\n\\left(1+\\frac{1}{\\ell(\\ell+1)}\\right)&\\text{if }\\nu_\\ell(N)<2\\nu_\\ell(f),\\\\\n\\left(1+\\frac{1}{\\ell}\\right)&\\text{if }\\nu_\\ell(N)>2\\nu_\\ell(f),\\\\\n\\left(1+\n\t\\frac{\\leg{-N_\\ell}{\\ell}\\ell+\\leg{-N_\\ell}{\\ell}-\\leg{N_\\ell}{\\ell}-1}{\\ell(\\ell^2-1)}\n\t\\right)\n\t&\\text{if }\\nu_\\ell(N)=2\\nu_\\ell(f).\n\\end{cases}\n\\end{align*}\nSubstituting this back into equation~\\eqref{separate n and f}, we have\n\\begin{equation}\\label{f sum remaining}\nK_0(N)\n=\\frac{2}{3}\n\t\\prod_{\\ell\\mid N}F_0(\\ell)\n\t\\prod_{\\substack{\\ell \\nmid N\\\\ \\ell\\ne 2}} F_1(\\ell)\n\t\\sum_{\\substack{f=1\\\\ (f,2)=1}}^{\\infty}\\strut^\\prime\n\t\t\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{\\varphi(f)f^2}\n\t\t\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\mid N}}\\frac{F_2(\\ell,f)}{F_0(\\ell)}\n\t\t\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\nmid N}}\\frac{F_2(\\ell,f)}{F_1(\\ell)}.\n\\end{equation}\nThe sum over $f$ may be factored as\n\\begin{equation*}\n\\begin{split}\n&\\sum_{\\substack{f=1\\\\ (f,2)=1}}^{\\infty}\\strut^\\prime\n\t\t\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{\\varphi(f)f^2}\n\t\t\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\mid N}}\\frac{F_2(\\ell,f)}{F_0(\\ell)}\n\t\t\\prod_{\\substack{\\ell\\mid f\\\\ \\ell\\nmid N}}\\frac{F_2(\\ell,f)}{F_1(\\ell)}\\\\\n&\\quad=\\prod_{\\ell\\mid N}\n\t\\left\\{1+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_0(\\ell)}\n\t\t\\right\\}\n\\prod_{\\substack{\\ell\\nmid N\\\\ \\ell\\ne 2}}\n\t\\left\\{1+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_1(\\ell)}\n\t\\right\\}.\n\\end{split}\n\\end{equation*}\nWhen $\\ell\\nmid 2N$, the factor simplifies as\n\\begin{equation*}\n\\begin{split}\n1+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_1(\\ell)}\n&=1+\\frac{\\ell C_N^{(\\ell)}(1,1,\\ell)F_2(\\ell,\\ell)}{F_1(\\ell)(\\ell-1)}\n\t\\sum_{\\alpha\\ge 1}\\frac{1}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\left(1+\\leg{N(N-1)^2}{\\ell}\\right)(\\ell^2+\\ell+1)}{(\\ell^2-1)(\\ell^3-1)F_1(\\ell)}\\\\\n&=1+\\frac{1+\\leg{N(N-1)^2}{\\ell}}{(\\ell^2-1)(\\ell-1)F_1(\\ell)}.\n\\end{split}\n\\end{equation*}\nWhen $\\nu_\\ell(N)$ is odd, the factor \nsimplifies as\n\\begin{equation*}\n\\begin{split}\n1&+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_0(\\ell)}\\\\\n\\quad&=1+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha\\ge 1}\n\t\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\n\t\\sum_{\\alpha=1}^{\\floor{\\nu_\\ell(N)\/2}}\n\t\\frac{\\ell^{\\alpha}\\left(1+\\frac{1}{\\ell}\\right)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{(\\ell+1)(1-\\ell^{1-\\nu_\\ell(N)})}{F_0(\\ell)(\\ell-1)(\\ell^2-1)}\\\\\n&=1+\\frac{1-\\ell^{1-\\nu_\\ell(N)}}{F_0(\\ell)(\\ell-1)^2}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}.\n\\end{split}\n\\end{equation*}\nWhen $\\nu_\\ell(N)$ positive, even, and $\\leg{N_\\ell}{\\ell}=-1$, the factor simplifies as\n\\begin{equation*}\n\\begin{split}\n1&+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_0(\\ell)}\\\\\n\\quad&=1+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha\\ge 1}\n\t\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell^2}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell\\#C_N^{(\\ell)}(1,1,\\ell^{\\nu_\\ell(N)\/2})F_2(\\ell,\\ell^{\\nu_\\ell(N)\/2})}\n\t{F_0(\\ell)(\\ell-1)\\ell^{3\\nu_\\ell(N)\/2}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell^2}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell^2-\\ell-\\leg{-1}{\\ell}}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell-\\leg{-1}{\\ell}}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell+\\leg{-N_\\ell}{\\ell}}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}.\n\\end{split}\n\\end{equation*}\nWhen $\\nu_\\ell(N)$ positive, even, and $\\leg{N_\\ell}{\\ell}=1$, the factor simplifies as\n\\begin{equation*}\n\\begin{split}\n1&+\\sum_{\\alpha\\ge 1}\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}\n\t\t{\\varphi(\\ell^\\alpha)\\ell^{2\\alpha}F_0(\\ell)}\n=1+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha\\ge 1}\n\t\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell\\#C_N^{(\\ell)}(1,1,\\ell^{\\nu_\\ell(N)\/2})F_2(\\ell,\\ell^{\\nu_\\ell(N)\/2})}\n\t{F_0(\\ell)(\\ell-1)\\ell^{3\\nu_\\ell(N)\/2}}\\\\\n&\\quad+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha=\\frac{\\nu_\\ell(N)}{2}+1}^\\infty\n\t\\frac{\\#C_N^{(\\ell)}(1,1,\\ell^\\alpha) F_2(\\ell,\\ell^\\alpha)}{\\ell^{3\\alpha}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell^2}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell(\\ell^2-1)+\\leg{-1}{\\ell}\\ell+\\leg{-1}{\\ell}-2}\n\t{F_0(\\ell)(\\ell-1)(\\ell^2-1)\\ell^{\\nu_\\ell(N)}}\\\\\n&\\quad+\\frac{\\ell}{F_0(\\ell)(\\ell-1)}\\sum_{\\alpha=\\frac{\\nu_\\ell(N)}{2}+1}^\\infty\n\t\\frac{2\\ell^{\\nu_\\ell(N)\/2}\\left(1+\\frac{1}{\\ell(\\ell+1)}\\right)}{\\ell^{3\\alpha}}\\\\\n\n\n\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell^2}{F_0(\\ell)\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\n\t+\\frac{\\ell^2-\\ell+\\leg{-1}{\\ell}}\n\t{F_0(\\ell)(\\ell-1)^2\\ell^{\\nu_\\ell(N)}}\\\\\n&=1+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell+\\leg{-N_\\ell}{\\ell}}{F_0(\\ell)(\\ell-1)^2\\ell^{\\nu_\\ell(N)}}.\n\\end{split}\n\\end{equation*}\n\nSubstituting this back into~\\eqref{f sum remaining}, we find that\n\\begin{equation*}\n\\begin{split}\nK_0(N)\n&=\\frac{2}{3}\n\t\\prod_{\\ell \\nmid 2N}\\left(F_1(\\ell)+\\frac{1+\\leg{N(N-1)^2}{\\ell}}{(\\ell^2-1)(\\ell-1)}\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\nmid\\nu_\\ell(N)}}\n\t\\left(F_0(\\ell)+\\frac{(\\ell^{\\nu_\\ell(N)}-\\ell)}{\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\right)\\\\\n&\\quad\\times\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\mid\\nu_\\ell(N)}}\n\t\\left(F_0(\\ell)\n\t+\\frac{\\ell^{\\nu_\\ell(N)}-\\ell+\\leg{-N_\\ell}{\\ell}}{\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\right)\\\\\n&=\\frac{2}{3}\n\t\\prod_{\\ell \\nmid 2N}\\left(\n\t1-\\frac{\\leg{N-1}{\\ell}^2\\left[\\ell+1-\\leg{N}{\\ell}\\right]+\\leg{N}{\\ell}}{(\\ell-1)(\\ell^2-1)}\n\t\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\nmid\\nu_\\ell(N)}}\n\t\\left(1+\\frac{\\ell^{\\nu_\\ell(N)+1}-\\ell^{\\nu_\\ell(N)}-\\ell}\n\t{\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\right)\\\\\n&\\quad\\times\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\mid\\nu_\\ell(N)}}\n\t\\left(1+\\frac{\\ell^{\\nu_\\ell(N)+1}-\\ell^{\\nu_\\ell(N)}-\\ell+\\leg{-N_\\ell}{\\ell}}\n\t{\\ell^{\\nu_\\ell(N)}(\\ell-1)^2}\\right)\\\\\n&=\\frac{N}{\\varphi(N)}\n\t\\prod_{\\ell \\nmid N}\\left(\n\t1-\\frac{\\leg{N-1}{\\ell}^2\\ell+1}{(\\ell-1)(\\ell^2-1)}\n\t\\right)\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\nmid\\nu_\\ell(N)}}\n\t\\left(1-\\frac{1}{\\ell^{\\nu_\\ell(N)}(\\ell-1)}\\right)\\\\\n&\\quad\\times\n\t\\prod_{\\substack{\\ell\\mid N\\\\ 2\\mid\\nu_\\ell(N)}}\n\t\\left(1-\\frac{\\ell-\\leg{-N_\\ell}{\\ell}}\n\t{\\ell^{\\nu_\\ell(N)+1}(\\ell-1)}\\right).\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\n\n\\section{Proof of Theorem~\\ref{main thm}}\\label{proof of main thm}\n\nWe are now ready to give the proof of our main result.\n\\begin{proof}[Proof of Theorem~\\ref{main thm}.]\nBy Proposition~\\ref{average order in terms of class numbers},\nwe see that Theorem~\\ref{main thm} follows if we show that\n\\begin{equation*}\n\\sum_{N^-<p< N^+}\\frac{H(D_N(p))}{p}\n=K(N)\\frac{N}{\\varphi(N)\\log N}+O\\left(\\frac{1}{(\\log N)^{1+\\gamma}}\\right).\n\\end{equation*}\nWe begin by dividing the interval $(N^-,N^+)$ into intervals of length\n$Y:=\\frac{\\sqrt N}{\\floor{(\\log N)^{\\gamma+2}}}$.  For each integer $k$ in\n$I:=[-2\\sqrt N\/Y, 2\\sqrt N\/Y)\\cap\\mathbb{Z}$,\nwe write $X=X_k:=N+1+kY$.\nThus,\n\\begin{equation}\\label{break into shorter averages}\n\\sum_{N^-<p< N^+}\\frac{H(D_N(p))}{p}\n=\\sum_{k\\in I}\\sum_{X_k<p\\le X_k+Y}\\frac{H(D_N(p))}{p}.\n\\end{equation}\n\n\n\nRecalling the definition of the Kronecker class number, the definition of $d_{N,f}(p)$, and the\nclass number formula\n(see~\\eqref{defn of K-class no},~\\eqref{defn of d}, and~\\eqref{class no formula}),\nwe have the identity\n\\begin{equation}\\label{analytic Kronecker class number formula}\nH(D_N(p))=\\frac{1}{2\\pi}\\sum_{\\substack{f^2|D_N(p)\\\\ d_{N,f}(p)\\equiv 0,1\\pmod 4}}\n\\sqrt{|d_{N,f}(p)|}L\\left(1,\\chi_{d_{N,f}(p)}\\right).\n\\end{equation}\nWe assume that $N$ is large enough so that there are only odd primes $p$ satisfying\nthe condition $N^-<p<N^+$.\nTherefore, since we assumed that $N$ is odd,\n\\begin{equation*}\nD_N(p)=(p+1-N)^2-4p\\equiv 1\\pmod 4,\n\\end{equation*}\nand it follows that there are only odd $f$ in the above sum and that\n$d_{N,f}(p)\\equiv 1\\pmod 4$ for each such $f$.\nHence, summing~\\eqref{analytic Kronecker class number formula} over all primes $p$\nin the range $X<p\\le X+Y$ and switching the order of summation, we have\n\\begin{equation}\\label{reduce to short average of L-values}\n\\begin{split}\n\\sum_{X<p\\le X+Y}\\frac{H(D_N(p))}{p}\n&=\\frac{1}{2\\pi}\\sum_{X<p\\le X+Y}\\frac{1}{p}\n\\sum_{f^2|D_N(p)}\n\\sqrt{|d_{N,f}(p)|}L\\left(1,\\chi_{d_{N,f}(p)}\\right)\\\\\n&=\\frac{1}{2\\pi}\\sum_{\\substack{f\\le 2\\sqrt{X+Y}\\\\ (f,2)=1}}\\frac{1}{f}\n\\sum_{\\substack{X<p\\le X+Y\\\\ f^2\\mid D_N(p)}}\n\\frac{\\sqrt{|D_N(p)|}}{p}L\\left(1,\\chi_{d_{N,f}(p)}\\right).\n\\end{split}\n\\end{equation}\n\nWe now change ``weights,\" approximating $\\frac{\\sqrt{|D_N(p)|}}{p}$ by\n$\\frac{\\sqrt{|D_N(X)|}\\log p}{N\\log N}$.  If $p$ is a prime in the interval $(X,X+Y]$,\nthen $p=X+O(Y)$, and hence\n\\begin{equation*}\nD_N(p)\n=D_N(X)+O(Y\\sqrt N).\n\\end{equation*}\nLet $X^*$ be the value minimizing the function $\\sqrt{|D_N(t)|}$ on the interval $[X,X+Y]$.\nSince it is also true that $p=N+O(\\sqrt N)$, we have that\n\\begin{equation*}\n\\left|\\frac{\\sqrt{|D_N(p)|}}{p}-\\frac{\\sqrt{|D_N(X)|}\\log p}{N\\log N}\\right|\n\\ll\\begin{cases}\n\\frac{\\sqrt{|D_N(X)|}}{N^{3\/2}}+\\frac{\\sqrt{Y}}{N^{3\/4}}&\\text{if }N^{\\pm}\\in [X,X+Y],\\\\\n\\frac{\\sqrt{|D_N(X)|}}{N^{3\/2}}+\\frac{Y}{N^{1\/2}\\sqrt{|D_N(X^*)|}}&\\text{otherwise.}\n\\end{cases}\n\\end{equation*}\nHence, by Theorem~\\ref{thmaverageSV} and Proposition~\\ref{product formula}, the right hand side\nof~\\eqref{reduce to short average of L-values} is equal to\n\\begin{equation}\\label{change weights}\n\\begin{split}\n\\frac{K(N)Y\\sqrt{|D_N(X)|}}{2\\pi \\varphi(N)\\log N}\n+\\begin{cases}\nO\\left(\\frac{Y\\sqrt{|D_N(X)|}}{N(\\log N)^{\\gamma+1}}\n\t+\\frac{Y\\sqrt{|D_N(X)|}\\log N}{N^{3\/2}}+\\frac{Y^{3\/2}\\log N}{N^{3\/4}}\\right)\n\t&\\text{ if }N^\\pm\\in [X,X+Y],\\\\\nO\\left(\n\\frac{Y\\sqrt{|D_N(X)|}}{N(\\log N)^{\\gamma+1}}\n\t+\\frac{Y\\sqrt{|D_N(X)|}\\log N}{N^{3\/2}}\n\t+\\frac{Y^2\\log N}{N^{1\/2}\\sqrt{|D_N(X^*)|}} \\right)\n\t&\\text{otherwise}.\n\\end{cases}\n\\end{split}\n\\end{equation}\nSince $D_N(X_k)=0$ for $k$ on the endpoints of the interval $[-2\\sqrt N\/Y,2\\sqrt N\/Y]\\supset I$,\nby the Euler-Maclaurin summation formula, we have\n\\begin{equation*}\n\\begin{split}\n\\sum_{k\\in I}\\sqrt{|D_N(X_k)|}\n&=\\int_{-2\\sqrt N\/Y}^{2\\sqrt N\/Y}\\sqrt{4N-(tY)^2}\\mathrm{d} t\n+O\\left(\\int_{-2\\sqrt N\/Y}^{2\\sqrt N\/Y}\\frac{\\left| tY^2\\right|}{\\sqrt{4N-(tY)^2}}\\mathrm{d} t\\right)\\\\\n&=\\frac{2\\pi N}{Y}\n+O\\left(\\sqrt N\\right),\n\\end{split}\n\\end{equation*}\nand furthermore,\n\\begin{equation*}\n\\sum_{k\\in I}\\frac{1}{\\sqrt{|D_N(X_k^*)|}}\n\\ll\\int_{-2\\sqrt N\/Y}^{2\\sqrt N\/Y}\\frac{\\mathrm{d} t}{\\sqrt{4N-(tY)^2}}\n=\\frac{1}{Y}\\int_{-2\\sqrt N}^{2\\sqrt N}\\frac{\\mathrm{d} u}{\\sqrt{4N-u^2}}\n=\\frac{\\pi}{Y}.\n\\end{equation*}\nWhence, summing~\\eqref{change weights} over $k\\in I$, we have\n\\begin{equation*}\n\\sum_{N^-<p< N^+}\\frac{H(D_N(p))}{p}\n=K(N)\\frac{N}{\\varphi(N)\\log N}\n\t+O\\left( \\frac{1}{(\\log N)^{\\gamma+1}}\n\t+\\frac{Y\\log N}{\\sqrt N}\n\t\\right).\n\\end{equation*}\n\\end{proof}\n\n\n\\section{Proofs of lemmas}\\label{proofs of lemmas}\n\n\nIn this section, we give the proofs of the technical lemmas needed in the rest of the paper.\n\n\\begin{lma}\\label{bound quad cong soln count}\nFor every positive integer $f$,\n\\begin{equation*}\n\\#\\{a\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(a)\\equiv 0\\pmod f\\}\\ll\\sqrt f.\n\\end{equation*}\n\\end{lma}\n\\begin{rmk}\nRecall that since this lemma was used in the proof of Proposition~\\ref{upper bound for H on avg},\nwe do not assume that $N$ is odd here.\n\\end{rmk}\n\\begin{proof}[Proof of Lemma~\\ref{bound quad cong soln count}.]\nFirst, we use the Chinese Remainder Theorem to write\n\\begin{equation*}\n\\#\\{a\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(a)\\equiv 0\\pmod{f}\\}\n=\\prod_{\\ell\\mid f}\\#\\{a\\in\\mathbb{Z}\/\\ell^{\\nu_\\ell(f)}\\mathbb{Z}: D_N(a)\\equiv 0\\pmod{\\ell^{\\nu_\\ell(f)}}\\}.\n\\end{equation*}\nWe will show that\n\\begin{equation}\\label{prime power mod bound}\n\\#\\{a\\in\\mathbb{Z}\/\\ell^{e}\\mathbb{Z}: D_N(a)\\equiv 0\\pmod{\\ell^{e}}\\}\n\\le\\begin{cases}\n\\max\\{\\ell^{\\floor{e\/2}},2\\ell^{(e-1)\/2}\\}&\\text{if }\\ell>2,\\\\\n\\max\\{\\ell^{\\floor{e\/2}},4\\ell^{(e-1)\/2}\\}&\\text{if }\\ell=2.\n\\end{cases}\n\\end{equation}\nFrom this, we readily deduce that\n\\begin{equation*}\n\\#\\{a\\in\\mathbb{Z}\/f\\mathbb{Z}: D_N(a)\\equiv 0\\pmod{f}\\}\\le 8\\sqrt f,\n\\end{equation*}\nwhich is a more precise result than stated in the lemma.\n\nWe now give the proof of~\\eqref{prime power mod bound}.\nSince\n\\begin{equation*}\nD_N(a)=a^2-2(N+1)a+(N-1)^2=(a-N-1)^2-4N,\n\\end{equation*}\nit suffices to consider the number of solutions to the congruence\n\\begin{equation}\\label{change of var}\nZ^2\\equiv 4N\\pmod{\\ell^e}.\n\\end{equation}\nSuppose $z$ is an integer solution to~\\eqref{change of var} and\nwrite $4N=\\ell^sN_0$ with $(\\ell,N_0)=1$.\nIf $s\\ge e$, it follows that $z\\equiv 0\\pmod{\\ell^{\\ceil{e\/2}}}$, and hence there are at most\n$\\ell^{\\floor{e\/2}}$ solutions to~\\eqref{change of var}.\nThus, we may assume that $s<e$ and write\n\\begin{equation*}\nz^2= \\ell^s(N_0+\\ell^{e-s}k)\n\\end{equation*}\nfor some integer $k$.\nSince $(\\ell,N_0)=1$, it follows that $s$ must be even.\nWriting $s=2s_0$, we see that $z=\\ell^{s_0}x$, where\n$x$ is an integer solution to the congruence\n\\begin{equation}\\label{reduced cong}\nX^2\\equiv N_0\\pmod{\\ell^{e-s}}.\n\\end{equation}\nSo, in particular, $z\\equiv\\ell^{s_0}x\\pmod{\\ell^{e-s_0}}$.\nSince $(\\ell,N_0)=1$, it is a classical exercise as in~\\cite[p.~98]{HW:1979} to show that\n\\begin{equation*}\n\\#\\{X\\in\\mathbb{Z}\/\\ell^{e-s}\\mathbb{Z}: X^2\\equiv N_0\\pmod{\\ell^{e-s}}\\}\n\\le\\begin{cases}\n2&\\text{if }\\ell>2,\\\\\n4&\\text{if }\\ell=2.\n\\end{cases}\n\\end{equation*}\nTherefore, there are at most $2\\ell^{e-(e-s_0)}=2\\ell^{s\/2}\\le 2\\ell^{(e-1)\/2}$ solutions\nto~\\eqref{change of var} when $\\ell$ is odd, and there are at most\n$4\\ell^{e-(e-s_0)}=4\\ell^{s\/2}\\le 4\\ell^{(e-1)\/2}$\nsolutions when $\\ell=2$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{complete K sum}]\nBy Lemma~\\ref{compute factors of c},\n$c_{N,f}(n)\\ll \\frac{n\\prod_{\\ell\\mid(f,n)}\\#C_N^{(\\ell)}(1,1,f)}{\\kappa_{2N}(n)}$,\nwhere for any positive integer $m$,\n$\\kappa_{m}(n)$ is the multiplicative function defined on the prime powers\nby~\\eqref{defn of kappa}.\nTherefore,\n\\begin{equation}\\label{diff bn K and truncated K}\n\\begin{split}\nK_0(N)&-\\sum_{\\substack{f\\le V\\\\ (f,2)=1}}\\frac{1}{f}\\sum_{n\\le U}\\frac{1}{n\\varphi(4nf^2)}\n\t\\sum_{\\substack{a\\in\\mathbb{Z}\/4n\\mathbb{Z}\\\\ a\\equiv 1\\pmod 4}}\\leg{a}{n}\\#C_N(a,n,f)\\\\\n&\\quad\\ll\\sum_{f\\le V}\\strut^\\prime\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n>U}\\frac{2\\varphi((n,f))c_{N,f}(n)}{(n,f)n\\varphi(4n)\n\t\\prod_{\\ell\\mid (n,f)}\\#C_N^{(\\ell)}(1,1,f)}\\\\\n\t&\\quad\\quad+\\sum_{f> V}\\strut^\\prime\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n\\ge 1}\\frac{2\\varphi((n,f))c_{N,f}(n)}{(n,f)n\\varphi(4n)\n\t\\prod_{\\ell\\mid (n,f)}\\#C_N^{(\\ell)}(1,1,f)}\\\\\n&\\quad\\ll\\sum_{\\substack{f\\le V\\\\ (f,2)=1}}\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n>U}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\\\\\n\t&\\quad\\quad+\\sum_{\\substack{f> V\\\\ (f,2)=1}}\n\t\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n\t\\sum_{n\\ge 1}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)},\n\\end{split}\n\\end{equation}\nwhere the primes on the sums on $f$ are meant to indicate that the sums are to be restricted\nto odd $f$ such that $\\#C_N^{(\\ell)}(1,1,f)\\ne 0$ for all primes $\\ell$ dividing $f$.\n\n\nIn~\\cite[Lemma 3.4]{DP:1999}, we find that\n\\begin{equation*}\n\\sum_{n>U}\\frac{1}{\\kappa_1(n)\\varphi(n)}\n\\sim\\frac{c_0}{\\sqrt U}\n\\end{equation*}\nfor some positive constant $c_0$.\nIn particular, this implies that the full sum converges.\nFrom this we obtain a crude bound for the tail of the sum over $n$\n\\begin{equation*}\n\\begin{split}\n\\sum_{n>U}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\n&=\\sum_{\\substack{km>U\\\\ (m,2N)=1\\\\ \\ell\\mid k\\Rightarrow\\ell\\mid 2N}}\n\t\\frac{1}{\\kappa_1(m)\\varphi(m)\\varphi(k)}\n=\\sum_{\\substack{k\\ge 1\\\\ \\ell\\mid k\\Rightarrow\\ell\\mid 2N}}\\frac{1}{\\varphi(k)}\n\t\\sum_{\\substack{m>U\/k\\\\ (m,2N)=1}}\\frac{1}{\\kappa_1(m)\\varphi(m)}\\\\\n&\\ll\\sum_{\\substack{k\\ge 1\\\\ \\ell\\mid k\\Rightarrow\\ell\\mid 2N}}\\frac{1}{\\varphi(k)}\n\t\\frac{\\sqrt k}{\\sqrt U}\n\\ll\\frac{1}{\\sqrt U}\\prod_{\\ell\\mid N}\\left(1+\\frac{\\ell}{(\\ell-1)(\\sqrt\\ell-1)}\\right)\\\\\n&=\\frac{1}{\\sqrt U}\\frac{N}{\\varphi(N)}\n\t\\prod_{\\ell\\mid N}\\left(1+\\frac{1}{\\sqrt\\ell(\\ell-1)}\\right)\\left(1+\\frac{1}{\\sqrt\\ell}\\right)\\\\\n&\\ll\\frac{1}{\\sqrt U}\\frac{N}{\\varphi(N)}\n\t\\prod_{\\ell\\mid N}\\left(1+\\frac{1}{\\sqrt\\ell}\\right).\n\\end{split}\n\\end{equation*}\nWe have already noted that $N\/\\varphi(N)\\ll\\log\\log N$.\nIt is a straightforward exercise as in~\\cite[p.~63]{MV:2007} to show that\n\\begin{equation*}\n\\prod_{\\ell\\mid N}\\left(1+\\frac{1}{\\sqrt\\ell}\\right)\n<\\exp\\left\\{O\\left(\\frac{\\sqrt{\\log N}}{\\log\\log N}\\right)\\right\\}.\n\\end{equation*}\nThus, we conclude that\n\\begin{equation}\\label{n tail}\n\\sum_{n>U}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\n\\ll\\frac{N^\\epsilon}{\\sqrt U}\n\\end{equation}\nfor any $\\epsilon>0$.\nFor the full sum over $n$, we need a sharper bound in the $N$-aspect, which we obtain\nby writing\n\\begin{equation}\\label{full n sum bound}\n\\begin{split}\n\\sum_{n\\ge 1}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\n&=\\prod_{\\ell\\mid 2N}\\left(1+\\frac{\\ell}{(\\ell-1)^2}\\right)\n\t\\sum_{\\substack{n\\ge 1\\\\ (n,2N)=1}}\\frac{1}{\\kappa_{2N}(n)\\varphi(n)}\\\\\n&\\le\\frac{2N}{\\varphi(2N)}\\prod_{\\ell\\mid 2N}\\left(1+\\frac{1}{\\ell(\\ell-1)}\\right)\n\t\\sum_{n\\ge 1}\\frac{1}{\\kappa_1(n)\\varphi(n)}\\\\\n&\\ll\\log\\log N.\t\n\\end{split}\n\\end{equation}\n\nFor any odd prime $\\ell$ dividing $f$, we obtain the bounds\n\\begin{equation*}\n\\#C_N^{(\\ell)}(1,1,f)\n\\le\\begin{cases}\n2\\ell^{\\nu_\\ell(N)\/2}&\\text{if }\\nu_\\ell(f)>\\nu_\\ell(N)\/2\\text{ and }2\\mid\\nu_\\ell(N),\\\\\n\\ell^{\\nu_\\ell(f)}&\\text{otherwise}\n\\end{cases}\n\\end{equation*}\nfrom Lemma~\\ref{counting solutions}.\nHowever, if $\\nu_\\ell(f)>\\nu_\\ell(N)\/2$ and $2\\mid\\nu_\\ell(N)$,\nit follows that $\\nu_\\ell(f)\\ge 1+\\nu_\\ell(N)\/2$, and hence\n\\begin{equation*}\n2\\ell^{\\nu_\\ell(N)\/2}\\le\\ell^{1+\\nu_\\ell(N)\/2}\\le\\ell^{\\nu_\\ell(f)}\n\\end{equation*}\nsince $\\ell>2$.\nTherefore, for every odd integer $f$, we have that\n\\begin{equation*}\n\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)\\le f,\n\\end{equation*}\nand hence\n\\begin{equation}\\label{f tail}\n\\sum_{\\substack{f> V\\\\ (f,2)=1}}\\frac{\\prod_{\\ell\\mid f}\\#C_N^{(\\ell)}(1,1,f)}{f^2\\varphi(f)}\n<\\sum_{f>V}\\frac{1}{f\\varphi(f)}\\ll\\frac{1}{V}.\n\\end{equation}\nSubstituting the bounds~\\eqref{n tail},~\\eqref{full n sum bound},\nand~\\eqref{f tail} into~\\eqref{diff bn K and truncated K}, the lemma follows.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{counting solutions}]\nUpon completing the square, we have that\n\\begin{equation*}\nz^2-2(N+1)z+(N-1)^2-af^2=(z-N-1)^2-(4N+af^2).\n\\end{equation*}\nSince $N$ is odd, the number of invertible solutions to the congruence\n\\begin{equation*}\n(z-N-1)^2\\equiv 4N+af^2\\pmod{2^\\alpha}\n\\end{equation*}\nis the same as the number of invertible solutions to the congruence\n$Z^2\\equiv 4N+af^2\\pmod{2^\\alpha}$.\nSince $4N+af^2\\equiv 4+a\\pmod{8}$, the computation of\n$C_N^{(2)}(a,n,f)$ is thus reduced to a standard exercise.\nSee~\\cite[p.~98]{HW:1979} for example.\n\nIf $\\ell\\nmid 4N+af^2$, then there are exactly $1+\\leg{4N+af^2}{\\ell}$\nsolutions to the congruence\n\\begin{equation}\\label{odd prime power cong}\n(z-N-1)^2\\equiv 4N+af^2\\pmod{\\ell^\\alpha}.\n\\end{equation}\nHowever,\nif $\\ell$ divides the constant term, $(N-1)^2-af^2$, then we have exactly one\ninvertible solution and exactly one noninvertible solution.\n\nIt remains to treat the case when $\\ell\\mid 4N+af^2$.\nWe write $4N+af^2=\\ell^sm$ with $(m,\\ell)=1$.\nFirst, we observe that any solution $z$ to~\\eqref{odd prime power cong} must\nsatisfy $z\\equiv N+1\\pmod{\\ell}$.  Therefore, if $\\ell\\mid N+1$, then there\nare no invertible solutions; if $\\ell\\nmid N+1$, then every solution is invertible.\nHence, we assume that $\\ell\\nmid N+1$.  Now, the number of invertible\nsolutions to~\\eqref{odd prime power cong} is equal to the number of\n(noninvertible) solutions to\n\\begin{equation}\\label{odd prime power cong with vanishing disc}\nZ^2\\equiv \\ell^sm\\pmod{\\ell^e}.\n\\end{equation}\nIf $s\\ge e$, then $Z$ is a solution if and only if\n$Z\\equiv 0\\pmod{\\ell^{\\ceil{e\/2}}}$.\nThere are exactly $\\ell^{e-\\ceil{e\/2}}=\\ell^{\\floor{e\/2}}$ such values for $Z$\nmodulo $\\ell^e$.\nNow, suppose that $0<s<e$.\nThen $Z$ is a solution to~\\eqref{odd prime power cong with vanishing disc} if and only if\n\\begin{equation*}\nZ^2=\\ell^sm+\\ell^e k=\\ell^s\\left(m+\\ell^{e-s}\\right)\n\\end{equation*}\nfor some integer $k$.\nSince $\\ell\\nmid m$ and $s<e$, we see that there can be no such $Z$ if\n$s$ is odd or if $\\leg{m}{\\ell}=-1$.\nThus, we assume that $s=2s_0$, $\\leg{m}{\\ell}=1$, and we write\n$r^2=m+\\ell^{e-s}$.\nUnder this assumption, there are exactly two (distinct modulo $\\ell^{e-s}$)\nsolutions, say $r_1$ and $r_2$, to the\ncongruence $r^2\\equiv m\\pmod{\\ell^{e-2s_0}}$.\nTherefore, if $Z$ is a solution to~\\eqref{odd prime power cong with vanishing disc},\nthen either $Z=\\ell^{s_0}(r_1+k_1\\ell^{e-s})$ for some integer $k_1$ or\n$Z=\\ell^{s_0}(r_2+k_2\\ell^{e-s})$ for some integer $k_2$.\nIn other words, $Z\\equiv\\ell^{s_0}r_1\\pmod{\\ell^{e-s_0}}$ or\n$Z\\equiv\\ell^{s_0}r_2\\pmod{\\ell^{e-s_0}}$.\nIt is not hard to check that if $Z$ satisfies either of these two conditions, then\n$Z$ is a solution to~\\eqref{odd prime power cong with vanishing disc}.\nThere are exactly $2\\ell^{e-(e-s_0)}=2\\ell^{s_0}$ such values\nfor $Z$ modulo $\\ell^e$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{compute factors of c}]\nIt is easily checked that $c_{N,f}(1)=1$.\nIf $n$ is odd, we observe that\n\\begin{equation*}\nc_{N,f}(n)=\\sum_{a\\in(\\mathbb{Z}\/n\\mathbb{Z})^*}\n\\leg{a}{n}\n\\prod_{\\ell\\mid n}\n\\#C_N^{(\\ell)}(a,n,f).\n\\end{equation*}\nNow, suppose that $(m,n)=1$.  It follows that at least one of $m$ and $n$ must be odd.\nWithout loss of generality,\nwe assume that $n$ is odd.  Choose integers $m_0$ and $n_0$ so that\n$4mm_0+nn_0=1$.\nThen\n\\begin{equation*}\n\\begin{split}\nc_{N,f}(n)c_{N,f}(m)&=\n\\sum_{a_1\\in(\\mathbb{Z}\/n\\mathbb{Z})^*}\n\\leg{a_1}{n}\n\\prod_{\\ell\\mid n}\n\\#C_N^{(\\ell)}(a_1,n,f)\\\\\n&\\quad\\times\n\\sum_{\\substack{a_2\\in(\\mathbb{Z}\/4m\\mathbb{Z})^*\\\\ a_2\\equiv 1\\pmod 4}}\n\\leg{a_2}{m}\n\\mathcal S_2(m,a_2)\n\\prod_{\\ell\\mid m'}\n\\#C_N^{(\\ell)}(a_2,m,f)\\\\\n&=\n\\sum_{\\substack{a_1\\in(\\mathbb{Z}\/n\\mathbb{Z})^*,\\\\ a_2\\in(\\mathbb{Z}\/4m\\mathbb{Z})^*\\\\ a_2\\equiv 1\\pmod 4}}\n\\leg{a_14mm_0+a_2nn_0}{nm}\n\\mathcal S_2(nm,a_14mm_0+a_2nn_0)\\\\\n&\\quad\\times\\prod_{\\ell\\mid nm'}\n\\#C_N^{(\\ell)}(a_14mm_0+a_2nn_0,nm,f)\\\\\n&=c_{N,f}(nm).\n\\end{split}\n\\end{equation*}\nHence $c_{N,f}(n)$ is multiplicative in $n$.\n\nBy~\\eqref{defn of S_2}, we find that\n\\begin{equation*}\nc_{N,f}(2^\\alpha)\n=\\sum_{\\substack{a\\in(\\mathbb{Z}\/2^{\\alpha+2}\\mathbb{Z})^*\\\\ a\\equiv 1\\pmod 4}}\n\\leg{a}{2^\\alpha}\\mathcal S_2(2^\\alpha,a)\n=2\\sum_{\\substack{a\\in(\\mathbb{Z}\/2^{\\alpha+2}\\mathbb{Z})^*\\\\ a\\equiv 5\\pmod 8}}\n\\leg{a}{2}^\\alpha\n=(-1)^\\alpha 2^{\\alpha}\n\\end{equation*}\nfor $\\alpha\\ge 1$.\n\nWe now consider the case when $\\ell^\\alpha$ is an odd prime power.\nIn view of Lemma~\\ref{counting solutions}, it is natural to split\n$c_{N,f}(\\ell^\\alpha)$ into two parts, writing\n\\begin{align*}\nc_{N,f}^{(1)}(\\ell^\\alpha)&:=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f),\\\\\nc_{N,f}^{(0)}(\\ell^\\alpha)&:=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\n\\end{align*}\nso that\n\\begin{equation}\\label{split c for odd ell}\nc_{N,f}(\\ell^\\alpha)\n=c_{N,f}^{(1)}(\\ell^\\alpha)\n+c_{N,f}^{(0)}(\\ell^\\alpha).\n\\end{equation}\n\nWe concentrate on $c_{N,f}^{(1)}(\\ell^\\alpha)$ first.\nApplying Lemma~\\ref{counting solutions}, we have\n\\begin{equation}\\label{nonvanishing disc part of c}\n\\begin{split}\nc_{N,f}^{(1)}(\\ell^\\alpha)\n&=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\nmid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)+\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\mid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\\\\\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\nmid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\left[1+\\leg{4N+af^2}{\\ell}\\right]\n+\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\mid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\\\\\n&=\\ell^{\\alpha-1}\\left\\{\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\nmid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\left[1+\\leg{4N+af^2}{\\ell}\\right]\n+\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\mid (N-1)^2-af^2}}\n\\leg{a}{\\ell}^\\alpha\n\\right\\}.\n\\end{split}\n\\end{equation}\nIn order to finish evaluating this sum, we split into cases.\nFirst, assume that $\\ell\\mid f$.  Then the sum defining $c_{N,f}^{(1)}(\\ell^\\alpha)$ is\nempty unless $\\ell\\nmid N$.\nIn this case, $\\ell\\mid (N-1)^2-af^2$ if and only if $\\ell\\mid N-1$.\nTherefore, if $\\ell\\mid f$ and $\\ell\\nmid N$, then\n\\begin{equation}\\label{c1 ell divides f}\n\\begin{split}\n\\frac{c_{N,f}^{(1)}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n&=\\begin{cases}\\displaystyle\n\\left[1+\\leg{N}{\\ell}\\right]\\sum_{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*}\\leg{a}{\\ell}^\\alpha\n\t&\\text{if }\\ell\\nmid N-1,\\\\\n\\displaystyle\n\\sum_{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*}\\leg{a}{\\ell}^\\alpha\t\n\t&\\text{if }\\ell\\mid N-1\n\\end{cases}\\\\\n&=\\begin{cases}\n(\\ell-1)\\left[1+\\leg{N}{\\ell}\\right]&\\text{if }2\\mid\\alpha\\text{ and }\\ell\\nmid N-1,\\\\\n\\ell -1&\\text{if }2\\mid\\alpha\\text{ and }\\ell\\mid N-1,\\\\\n0&\\text{if }2\\nmid\\alpha\n\\end{cases}\\\\\n&=\\begin{cases}\n(\\ell-1)\\left(1+\\leg{N}{\\ell}\\leg{N-1}{\\ell}^2\\right)\n\t&\\text{if }2\\mid\\alpha,\\\\\n0&\\text{if }2\\nmid\\alpha\n\\end{cases}\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\begin{cases}\n(\\ell-1)&\\text{if }2\\mid\\alpha,\\\\\n0&\\text{if }2\\nmid\\alpha\n\\end{cases}\n\\end{split}\n\\end{equation}\nas $\\#C_N^{(\\ell)}(1,1,f)=\\left(1+\\leg{N}{\\ell}\\leg{N-1}{\\ell}^2\\right)$ in this case.\n\nNow, suppose that $\\ell\\nmid f$.\nUnder this assumption, we note that $\\leg{a}{\\ell}=\\leg{af^2}{\\ell}$.\nPicking up with equation~\\eqref{nonvanishing disc part of c} and dividing through\nby $\\ell^{\\alpha-1}$, we have\n\\begin{equation}\\label{make c1 sum full}\n\\begin{split}\n\\frac{c_{N,f}^{(1)}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n&=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\nmid (N-1)^2-af^2}}\n\\leg{af^2}{\\ell}^\\alpha\n\\left[1+\\leg{4N+af^2}{\\ell}\\right]\n+\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+af^2\\\\ \\ell\\mid (N-1)^2-af^2}}\n\\leg{af^2}{\\ell}^\\alpha\\\\\n&=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+a\\\\ \\ell\\nmid (N-1)^2-a}}\n\\leg{a}{\\ell}^\\alpha\n\\left[1+\\leg{4N+a}{\\ell}\\right]\n+\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell\\mathbb{Z})^*\\\\ \\ell\\nmid 4N+a\\\\ \\ell\\mid (N-1)^2-a}}\n\\leg{a}{\\ell}^\\alpha\\\\\n&=\n\\sum_{\\substack{a\\in\\mathbb{Z}\/\\ell\\mathbb{Z}\\\\ a\\not\\equiv -4N\\pmod\\ell}}\n\\leg{a}{\\ell}^\\alpha\n+\\sum_{\\substack{a\\in\\mathbb{Z}\/\\ell\\mathbb{Z}\\\\  a\\not\\equiv (N-1)^2\\pmod\\ell}}\n\\leg{a}{\\ell}^\\alpha\\leg{4N+a}{\\ell}\\\\\n&=\\left[\\sum_{b=1}^\\ell\\left(1+\\leg{b}{\\ell}\\right)\\leg{b-4N}{\\ell}^\\alpha\\right]\n-\\leg{-N}{\\ell}^\\alpha-\\leg{N^2-1}{\\ell}^{2}\n\\end{split}\n\\end{equation}\nupon completing the sums and making the change of variables $b=4N+a$.\nOne easily computes that\n\\begin{equation*}\n\\sum_{b=1}^\\ell\\left(1+\\leg{b}{\\ell}\\right)\\leg{b-4N}{\\ell}^\\alpha\n=\\begin{cases}\n\\ell-1 &\\text{if }\\ell\\mid N,\\\\\n\\ell-1-\\leg{N}{\\ell} &\\text{if }\\ell\\nmid N\\text{ and }2\\mid\\alpha,\\\\\n-1 &\\text{if }\\ell\\nmid N\\text{ and }2\\nmid\\alpha.\n\\end{cases}\n\\end{equation*}\nFor example, the third case is Exercise 1(b) of~\\cite[p.~301]{MV:2007}.\nTherefore,\n\\begin{equation*}\n\\frac{c_{N,f}^{(1)}(\\ell^\\alpha)}{\\ell^{\\alpha-1}}\n=\\begin{cases}\n\\ell-2\n\t&\\text{if } \\ell\\mid N\\text{ and }\\ell\\nmid f,\\\\\n\\ell-2-\\leg{N}{\\ell}-\\leg{N^2-1}{\\ell}^2\n\t&\\text{if }\\ell\\nmid Nf\\text{ and }2\\mid\\alpha,\\\\\n-1-\\leg{-N}{\\ell}-\\leg{N^2-1}{\\ell}^2\n\t&\\text{if }\\ell\\nmid Nf\\text{ and }2\\nmid\\alpha.\n\\end{cases}\n\\end{equation*}\n\n\nIt remains to compute $c_{N,f}^{(0)}(\\ell^\\alpha)$.  We first consider the case when $\\ell\\nmid f$.\nNote that the sum defining $c_{N,f}^{(0)}(\\ell^\\alpha)$ is empty unless $\\ell\\nmid N$ as well.\nThus, under this assumption, we have that\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{af^2}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\\\\\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+a}}\n\\leg{a}{\\ell}^\\alpha\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,1)\\\\\n&=\\leg{-4N}{\\ell}^\\alpha\n\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+a}}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,1).\n\\end{split}\n\\end{equation*}\nIn order to evaluate this last sum, for each $a\\in(\\mathbb{Z}\/\\ell^{\\alpha}\\mathbb{Z})^*$,\n we choose an integer representative in the range $-4N< a\\le\\ell^\\alpha-4N$.  This ensures that\n$0\\le\\nu_\\ell(4N+a)\\le\\alpha$.\nThere is exactly one such choice of $a$ so that $\\nu_\\ell(4N+a)=\\alpha$, namely\n$a=\\ell^\\alpha-4N$.\nFor $1\\le t\\le\\alpha-1$,  the number of such $a$ with $\\nu_\\ell(4N+a)=t$ is\n$(\\ell-1)\\ell^{\\alpha-t-1}$, but for only half of those values is\n$\\leg{(4N+a)\/\\ell^t}{\\ell}=1$.  Perhaps, the easiest way to see this is to consider the\nbase-$\\ell$ expansion of $4N+a$ for each $a$ in this range.\nTherefore, applying Lemma~\\ref{counting solutions} again, we have\n\\begin{equation*}\n\\begin{split}\n\\sum_{t=1}^\\alpha\n\\sum_{\\substack{0< 4N+a\\le\\ell^\\alpha\\\\ \\nu_\\ell(4N+a)=t}}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,1)\n&=\\leg{N+1}{\\ell}^2\\ell^{\\floor{\\alpha\/2}}\n+\\sum_{t=1}^{\\floor{(\\alpha-1)\/2}}\n\\sum_{\\substack{0<4N+a<\\ell^\\alpha\\\\ \\nu_\\ell(4N+a)=2t}}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,1)\\\\\n&=\\leg{N+1}{\\ell}^2\\ell^{\\floor{\\alpha\/2}}\n+\\sum_{t=1}^{\\floor{(\\alpha-1)\/2}}\\frac{(\\ell-1)\\ell^{\\alpha-2t-1}}{2}\n\t2\\leg{N+1}{\\ell}^2\\ell^t\\\\\n&=\\leg{N+1}{\\ell}^2\\ell^{\\floor{\\alpha\/2}}\n\t+\\leg{N+1}{\\ell}^2\\ell^{\\alpha-1}\\left(1-\\ell^{-\\floor{(\\alpha-1)\/2}}\\right)\\\\\n&=\\leg{N+1}{\\ell}^2\\left(\n\\ell^{\\floor{\\alpha\/2}}+\n\\ell^{\\alpha-1}\\left(1-\\ell^{-\\floor{(\\alpha-1)\/2}}\\right)\n\\right)\\\\\n&=\\leg{N+1}{\\ell}^2\\ell^{\\alpha-1}.\n\\end{split}\n\\end{equation*}\nTherefore, if $\\ell\\nmid fN$, then\n\\begin{equation*}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n=\\leg{-N}{\\ell}^\\alpha\\leg{N+1}{\\ell}^2\\ell^{\\alpha-1}.\n\\end{equation*}\n\nWe now compute $c_{N,f}^{(0)}(\\ell^\\alpha)$ in the case that $\\ell\\mid f$.\nNote that, in this case, the sum defining $c_{N,f}^{(0)}(\\ell^\\alpha)$ is empty unless\n$\\ell\\mid N$.\nNote that $\\nu_\\ell(4N+af^2)\\ge\\min\\{\\nu_\\ell(N),2\\nu_\\ell(f)\\}$ with equality holding if\n$\\nu_\\ell(N)\\ne 2\\nu_\\ell(f)$.\n\nFirst, suppose that $2\\nu_\\ell(f)<\\nu_\\ell(N)$.\nNoting that $e=\\nu_\\ell(4\\ell^\\alpha f^2)>2\\nu_\\ell(f)$,\nby Lemma~\\ref{counting solutions} we have that\n\\begin{equation*}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\n=2\\ell^{\\nu_\\ell(f)}=2\\#C_N^{(\\ell)}(1,1,f)\n\\end{equation*}\nif and only if $\\leg{a}{\\ell}=\\leg{(4N+af^2)\/\\ell^{2\\nu_\\ell(f)}}{\\ell}=1$.\nHence,\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\\\\\n&=2\\#C_N^{(\\ell)}(1,1,f)\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\leg{a}{\\ell}=1}}\\leg{a}{\\ell}^\\alpha\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\ell^{\\alpha-1}(\\ell-1)\n\\end{split}\n\\end{equation*}\nif $1< 2\\nu_\\ell(f)<\\nu_\\ell(N)$.\n\nNow, suppose that $\\nu_\\ell(N)<2\\nu_\\ell(f)$. Since $e=\\nu_\\ell(4\\ell^\\alpha f^2)>\\nu_\\ell(N)$,\nby Lemma~\\ref{counting solutions}, we have that\n\\begin{equation*}\n\\begin{split}\n\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\n&=\\begin{cases}\n2\\ell^{\\nu_\\ell(N)\/2}&\\text{if } 2\\mid\\nu_\\ell(N)\\text{ and }\\leg{N_\\ell}{\\ell}=1,\\\\\n0&\\text{otherwise}\n\\end{cases}\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\n\\end{split}\n\\end{equation*}\nfor every $a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*$.\nHence,\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\sum_{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*}\n\\leg{a}{\\ell}^\\alpha\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\begin{cases}\n\\ell^{\\alpha-1}(\\ell-1)&\\text{if }2\\mid \\alpha,\\\\\n0&\\text{if }2\\nmid \\alpha\n\\end{cases}\n\\end{split}\n\\end{equation*}\nif $1\\le\\nu_\\ell(N)<2\\nu_\\ell(f)$.\n\nFinally, consider the case when $2\\nu_\\ell(f)=\\nu_\\ell(N)$.\nLet $r=\\nu_\\ell(f)$ and $s=\\nu_\\ell(N)$ and write\n$f=\\ell^rf_\\ell$ and $N=\\ell^sN_\\ell$ with $(\\ell,f_\\ell N_\\ell)=1$.\nThen\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\\sum_{\\substack{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*\\\\ \\ell\\mid 4N+af^2}}\n\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,f)\n=\\sum_{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*}\n\\leg{af_\\ell^2}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,\\ell^rf_\\ell)\\\\\n&=\\sum_{a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*}\n\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,\\ell^r).\n\\end{split}\n\\end{equation*}\nTo evaluate this last sum, for each $a\\in(\\mathbb{Z}\/\\ell^\\alpha\\mathbb{Z})^*$, we choose an\ninteger representative in the range $-4N_\\ell< a\\le\\ell^\\alpha-4N_\\ell$.\nThis ensures that $0\\le\\nu_\\ell(4N_\\ell+a)\\le\\alpha$, and hence that\n$2r\\le\\nu_\\ell(4N+a\\ell^{2r})\\le2r+\\alpha$.\nSimilar to before, there is exactly one choice of $a$ such that\n$\\nu_\\ell(4N_\\ell+a)=\\alpha$, namely $a=\\ell^\\alpha-4N_\\ell$.\nFor $1\\le t\\le\\alpha-1$, there are\n$(\\ell-1)\\ell^{\\alpha-t-1}$ choices with $\\nu_\\ell(4N_\\ell+a)=t$, but for only half of\nthose values is $\\leg{(4N_\\ell+a)\/\\ell^t}{\\ell}=1$.\nNote that if $\\ell\\mid 4N_\\ell+a$, then $\\leg{a}{\\ell}=\\leg{-N_\\ell}{\\ell}$.\nTherefore, if $1<\\nu_\\ell(N)=2\\nu_\\ell(f)$, then\n\\begin{equation*}\n\\begin{split}\nc_{N,f}^{(0)}(\\ell^\\alpha)\n&=\\sum_{t=0}^\\alpha\\sum_{\\substack{0< 4N_\\ell+a\\le \\ell^\\alpha\\\\ \\nu_\\ell(4N_\\ell+a)=t}}\n\t\\leg{a}{\\ell}^\\alpha\\#C_N^{(\\ell)}(a,\\ell^\\alpha,\\ell^r)\\\\\n&=\\leg{-N_\\ell}{\\ell}^\\alpha\\ell^{\\floor{(2r+\\alpha)\/2}}\n\t+\\sum_{t=1}^{\\floor{(\\alpha-1)\/2}}\n\t\\leg{-N_\\ell}{\\ell}^\\alpha\\frac{(\\ell-1)\\ell^{\\alpha-2t-1}}{2}2\\ell^{r+t}\\\\\n&\\quad+\\sum_{\\substack{0<4N_\\ell+a<\\ell^\\alpha\\\\ \\leg{4N_\\ell+a}{\\ell}=1}}\n\t\\leg{a}{\\ell}^\\alpha 2\\ell^r\\\\\n&=\\leg{-N_\\ell}{\\ell}^\\alpha\\ell^{r+\\floor{\\alpha\/2}}\n\t+\\leg{-N_\\ell}{\\ell}^\\alpha\\ell^{\\alpha-1+r}\\left(1-\\ell^{-\\floor{(\\alpha-1)\/2}}\\right)\\\\\n&\\quad+\\ell^{r+\\alpha-1}\n\t\\sum_{a\\in\\mathbb{Z}\/\\ell\\mathbb{Z}}\\leg{a}{\\ell}^\\alpha\\left(1+\\leg{4N_\\ell+a}{\\ell}\\right)\\\\\n&=\\leg{-N_\\ell}{\\ell}\\ell^r\\left[\\ell^{\\floor{\\alpha\/2}}\n\t+\\ell^{\\alpha-1}\\left(1-\\ell^{-\\floor{(\\alpha-1)\/2}}\\right)\\right]\\\\\n&\\quad+\\ell^{r+\\alpha-1}\n\t\\sum_{b\\in\\mathbb{Z}\/\\ell\\mathbb{Z}}\\left(1+\\leg{b}{\\ell}\\right)\\leg{b-4N_\\ell}{\\ell}^\\alpha\\\\\n&=\n\\ell^{r+\\alpha-1}\n\\begin{cases}\n\\leg{-N_\\ell}{\\ell}+\\ell-1-\\leg{N_\\ell}{\\ell}&\\text{if }2\\mid\\alpha,\\\\\n\\leg{-N_\\ell}{\\ell}-1&\\text{if }2\\nmid\\alpha\n\\end{cases}\\\\\n&=\\#C_N^{(\\ell)}(1,1,f)\\ell^{\\alpha-1}\n\\begin{cases}\n\\leg{-N_\\ell}{\\ell}+\\ell-1-\\leg{N_\\ell}{\\ell}&\\text{if }2\\mid\\alpha,\\\\\n\\leg{-N_\\ell}{\\ell}-1&\\text{if }2\\nmid\\alpha.\n\\end{cases}\n\\end{split}\n\\end{equation*}\nThe lemma now follows by combining our computations for $c_{N,f}^{(0)}(\\ell^\\alpha)$\nand $c_{N,f}^{(1)}(\\ell^\\alpha)$.\n\n\\end{proof}\n\n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}