On the mean value of \(L\)-functions with the weight of character sums
type: Item
instance of: scholarly article
title: On the mean value of \(L\)-functions with the weight of character sums
zbMATH Open document ID: 1181.11054
published in: Journal of Number Theory
publication date: 2009-06-17T00:00:00Z
review text: Let \(q\geq 2\) be an integer, \(\chi\) be a Dirichlet character modulo \(q\), and \(L(s,\chi)\) be the Dirichlet \(L\)-function corresponding to \(\chi\). The author employs analytic methods to study the mean value properties of \[ \sum_{_{\substack{ \chi\mod{q} \\ \chi \neq \chi_0 }}} \left|\sum_{n \leq N} \chi(n)\right|^2 \cdot |L(1,\chi)|^2 \] and establishes the following mean value result: If \(q\) is an integer greater than 2, then for any real number \(N\) with \(1<N<\sqrt{q}\) \[ \begin{aligned}{ \sum_{\substack{\chi\mod{q} \\ \chi\neq\chi_0}} \left|\sum_{n\leq N} \chi(n)\right|^2 \cdot |L(1,\chi)|^2 &={\pi^2 \phi^2(q) N \over 6q} \prod_{p\mid q} \left(1 - {1 \over p^2}\right) + {\pi^2 \phi^2(q) N \over 3q \zeta(3)} \cr &\quad \times\prod_{p\mid q} \left(1 - {1 \over p^2 + p + 1}\right) \mathop{\sum_{m=1}^{\infty}\!\!' \sum_{n=1}^{\infty}\!'}_{(m+n,q) = 1} \frac{1}{mn(m + n)} \cr &\quad + O(\phi(q) 2^{\omega(q)} \ln^2 q) + O(N^3 \ln^2 q), }\end{aligned} \] where \(\sum_{\chi\neq\chi_0}\) is the summation over all non-principal Dirichlet characters modulo \(q\), \(\prod_{p\mid q}\) is the product over all distinct prime divisors of \(q\), \(\sum_{m=1}^{'\infty}\) is the summation over all \(m\) that are relatively prime to \(q\), \(\phi(q)\) is the Euler totient function, \(\omega(q)\) is the number of distinct prime divisors of \(q\), and \(\zeta(s)\) is the Riemann zeta-function. \quad The author also proves the following two corollaries. First, if \(p\) is a prime greater than 2, then for any real \(N\) with \(1<N<\sqrt{p}\) \[ \begin{aligned}{ \sum_{\substack{\chi\mod{p} \\ \chi\neq\chi_0}} \left|\sum_{n\leq N} \chi(n)\right|^2 \cdot |L(1,\chi)|^2 &={\pi^2 p N \over 6} \left(1 + {2 \over \zeta(3)}\sum_{m=1}^{\infty} {1 \over m^2} \sum_{k=1}^{m} {1 \over k}\right) \cr &\quad + O(p\ln^2 p + N^3\ln^2 p). }\end{aligned} \] Moreover, for any real number \(N\) with \(p^{\epsilon}<N<p^{\frac{1}{2}-\varepsilon}\) (\(\varepsilon>0\) fixed) \[ {1 \over p} \sum_{_{…
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 5565660
zbMATH Keywords: Dirichlet character
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: Hybrid mean value
author: Wenpeng Zhang
reviewed by: Andrew Ledoan
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.11.008
OpenAlex ID: W1964782224
cites work: Q4101884
cites work: On the 2kth power mean of the character sums over short intervals
cites work: On the \(2k\)-th power mean of inversion of \(L\)-functions with the weight of the Gauss sum
cites work: On the \(2k\)-th power mean of Dirichlet \(L\)-functions with the weight of general Kloosterman sums
DOI: 10.1016/J.JNT.2007.11.008
description: scientific article; zbMATH DE number 5565660