Explicit formulae for the mean value of products of values of Dirichlet $L$-functions at positive integers

TL;DR

This paper derives explicit formulas for the average values of products of Dirichlet $L$-functions at positive integers, enhancing understanding of their mean behaviors and correlations.

Contribution

It provides the first explicit formulas for mean values of products of Dirichlet $L$-functions at positive integers, including complex products involving multiple characters.

Findings

Explicit formulas for mean values of $|L(m,\chi)|^2$ over characters.

Extension of formulas to products of multiple $L$-functions and their conjugates.

Methodology adaptable to various products of Dirichlet $L$-functions.

Abstract

Let $m\ge 1$ be a rational integer. We give an explicit formula for the mean value $$\frac{2}{\phi(f)}\sum_{\chi (-1)=(-1)^m}\vert L(m,\chi )\vert^2,$$ where $\chi$ ranges over the $\phi (f)/2$ Dirichlet characters modulo $f>2$ with the same parity as $m$. We then adapt our proof to obtain explicit means values for products of the form $L(m_1,\chi_1)\cdots L(m_{n-1},\chi_{n-1})\overline{L(m_n,\chi_1\cdots\chi_{n-1})}$.