Christopher Atherfold

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

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 explic...