On the $\beta =2$ Partition function for Dirichlet $L$-functions in the $q$-aspect

Abstract

We study the $\beta =2$ partition function ${\int }_{|h|\le {\log }^{\theta }(q)/2}|L(1/2+ih,\chi ){|}^{2}dh$ for typical Dirichlet characters modulo a large prime $q$ and $\theta \in (-1/2,0]$ motivated by a $q$-analogue of the Saksman--Webb conjectures. When $\theta <0$, we use Harper's randomisation argument to introduce explicit conditioning to recover moment upper bounds consistent with critical normalisation predicted there. As an application, we prove that for $q(1-o(1))$ Dirichlet characters modulo $q$, ${\max }_{|h|\le 1/2}|L(1/2+ih,\chi )|\ll \frac{\log (q)}{(\log \log (q){)}^{3/4+o(1)}}$, establishing an upper bound matching the predictions of the $q$-analogue of the Fyodorov--Hiary--Keating conjectures up to second order.

34 pages. Comments welcome