On the Moments of Certain Families of Dirichlet $L$-functions

TL;DR

This paper derives asymptotic formulas for the moments of certain families of Dirichlet L-functions, focusing on the effects of restricting to specific arithmetic progressions, which introduces complex non-diagonal terms.

Contribution

It provides detailed analysis of non-diagonal contributions in the moments of Dirichlet L-functions within arithmetic progressions, extending prior work by Soundararajan.

Findings

Derived asymptotic formulas for second moments

Analyzed non-diagonal term contributions

Extended understanding of L-function families in arithmetic progressions

Abstract

In this paper we address the problem of computing asymptotic formulae for the expected values and second moments of central values of primitive Dirichlet $L$-functions $L(1/2,\chi_{8d}\otimes\psi)$ when $\psi$ is a fixed even primitive non-quadratic character of odd modulus $q$, $\chi_{8d}$ is a primitive quadratic character, $d\equiv h\pmod r$ is odd and squarefree and $r\equiv0\pmod q$ is even. Restricting to these arithmetic progressions ensures that the resulting sets of $L$-functions each form a ``family of primitive $L$-functions" in the specific sense defined by Conrey, Farmer, Keating, Rubinstein and Snaith. Soundararajan had previously computed these quantities without restricting them to arithmetic progressions. It turns out that the restriction to arithmetic progressions introduces non-diagonal terms that require significantly more detailed analysis which we carry on in this paper.