Twisted $2k$th moments of primitive Dirichlet $L$-functions: beyond the diagonal

TL;DR

This paper rigorously analyzes the twisted 2kth moments of primitive Dirichlet L-functions for all even primitive characters with conductors up to Q, providing evidence beyond diagonal terms and confirming conjectured asymptotics.

Contribution

It offers the first rigorous proof of the asymptotic formula for twisted moments beyond diagonal terms for this family of L-functions under the Generalized Lindelöf Hypothesis.

Findings

Asymptotic formula matches conjectures by Conrey et al.

Provides rigorous evidence beyond diagonal contributions.

Validates predictions for the 2kth moments of Dirichlet L-functions.

Abstract

We study the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to $\infty$. For an arbitrary positive integer $k$, we approximate the twisted $2k$th moment of this family by using Dirichlet polynomial approximations of $L^k(s,\chi)$ of length $X$, with $Q<X<Q^2$. Assuming the Generalized Lindel\"{o}f Hypothesis, we prove an asymptotic formula for these approximations of the twisted moments. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith for this family of $L$-functions, and provides the first rigorous evidence beyond the diagonal terms for their conjectured asymptotic formula for the general $2k$th moment of this family.