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

Abstract

We study the $\beta=2$ partition function $\int_{|h| \leq \log^{\theta}(q)/2}|L(1/2+ih,\chi)|^2dh$ 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 $q$, $\max_{|h| \leq 1/2}|L(1/2+ih,\chi)|\ll \frac{\log(q)}{(\log\log(q))^{3/4+o(1)}}$, establishing an upper bound matching the predictions of the $q$-analogue of the Fyodorov--Hiary--Keating conjectures up to second order.