On the mean value of \(L\)-functions with the weight of character sums
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) \[ {1 \over p} \sum_{_{\substack{ \chi\mod{p} \\ \chi\neq\chi_0 }}} \left|\sum_{n\leq N} \chi(n)\right|^2 \cdot |L(1,\chi)|^2 \sim {\pi^2 N \over 6} \left(1 + {2 \over \zeta(3)} \sum_{m=1}^{\infty} {1 \over m^2} \sum_{k=1}^{\infty} {1 \over k}\right), \quad p\to +\infty. \]
scientific article; zbMATH DE number 5565660