The sixth moment of Dirichlet L-functions at the central point

TL;DR

This paper proves an asymptotic formula for the sixth moment of Dirichlet L-functions at the central point, removing the need for additional averaging and overcoming new technical challenges.

Contribution

It removes the extra short t-averaging in the sixth moment problem, establishing a precise asymptotic for the original conjecture.

Findings

Established an asymptotic for the sixth moment without extraneous averaging.

Solved technical difficulties from unbalanced sums in the analysis.

Extended previous bounds to exact asymptotics for the moment.

Abstract

In 1970, Huxley obtained a sharp upper bound for the sixth moment of Dirichlet $L$-functions at the central point, averaged over primitive characters $\chi$ modulo $q$ and all moduli $q \leq Q$. In 2007, as an application of their ``asymptotic large sieve'', Conrey, Iwaniec and Soundararajan showed that when an additional short $t$-averaging is introduced into the problem, an asymptotic can be obtained. In this paper we show that this extraneous averaging can be removed, and we thus obtain an asymptotic for the original moment problem considered by Huxley. The main new difficulty in our work is the appearance of certain challenging ``unbalanced'' sums that arise as soon as the $t$-aspect averaging is removed.