Moments of the Hurwitz zeta Function on the Critical Line

Anurag Sahay

Published 2021-03-25Version 1

We study the moments $M_k(T;\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$ of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line, $s = 1/2 + it$. We conjecture, in analogy with the Riemann zeta function, that $M_k(T;\alpha) \sim c_k(\alpha) T (\log T)^{k^2}$. In the case of $\alpha\in\mathbb{Q}$, we use heuristics from analytic number theory and random matrix theory to compute $c_k(\alpha)$. In the process, we investigate moments of products of Dirichlet $L$-functions on the critical line. We provide several pieces of evidence for our conjectures, in particular by proving some of them for the cases $k = 1,2$ and $\alpha \in \mathbb{Q}$.