The eighth moment of Dirichlet $L$-functions

TL;DR

This paper establishes an asymptotic formula for the eighth moment of Dirichlet L-functions averaged over characters and moduli, conditional on GRH, confirming conjectured constants and extending to GL(2) twists.

Contribution

It provides the first asymptotic for the eighth moment of Dirichlet L-functions and the fourth moment for GL(2) twists, matching conjectured constants.

Findings

Confirmed the conjectured constant 24024 in the eighth moment

Derived the asymptotic for the eighth moment of Dirichlet L-functions

Extended results to the fourth moment of GL(2) twists

Abstract

We prove an asymptotic for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $\chi$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2) L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24024 appears as a factor in the leading order term of the eighth moment.