The sixth power moment of Dirichlet L-functions

TL;DR

This paper establishes a precise formula for the sixth moment of Dirichlet L-functions averaged over moduli and characters, confirming predictions from random matrix theory and matching the Riemann zeta-function's sixth moment.

Contribution

It provides the first rigorous proof with power savings for the sixth moment of Dirichlet L-functions, aligning with theoretical predictions.

Findings

The formula matches predictions from random matrix theory.

The leading constant 42 appears as predicted.

The result includes power savings in the average.

Abstract

We prove a formula, with power savings, for the sixth moment of Dirichlet L-functions averaged over moduli $q$, over primitive characters $\chi$ modulo $q$, and over the critical line. Our formula agrees precisely with predictions motivated by random matrix theory. In particular, the constant 42 appears as a factor in the leading order term, exactly as is predicted for the sixth moment of the Riemann zeta-function.