On Some Weighted Average Values of L-functions

TL;DR

This paper extends W. Zhang's 2008 result on weighted averages of L-functions, showing the formula's validity over a broader range of N relative to q, enhancing understanding of L-function behavior.

Contribution

The paper generalizes Zhang's formula to a wider range of N, specifically from q^{rac{1}{2}- ext{epsilon}} to q^{1- ext{epsilon}}, broadening the applicability of the weighted average results.

Findings

The formula holds for q^{ ext{epsilon}} extless N extless q^{1- ext{epsilon}}.

The range of N for which the formula is valid is significantly extended.

The result provides deeper insight into the distribution of L-values for characters modulo q.

Abstract

Let $q\ge 2$ and $N\ge 1$ be integers. W. Zhang (2008) has shown that for any fixed $\epsilon> 0$, and $q^{\epsilon} \le N \le q^{1/2 -\epsilon}$, $$ \sum_{\chi \ne \chi_0} |\sum_{n=1}^N \chi(n)|^2 |L(1, \chi)|^2 = (1 + o(1)) \alpha_q q N $$ where the sum is take over all nonprincipal characters $\chi$ modulo $q$, $L(s, \chi)$ is the $L$-functions $L(1, \chi)$ corresponding to $\chi$ and $\alpha_q = q^{o(1)}$ is some explicit function of $q$. Here we show that the same formula holds in the range $q^{\epsilon} \le N \le q^{1 -\epsilon}$.