Moments of the Hurwitz zeta function on the critical line

Abstract

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, $s=1/2+it$ with a rational shift $\alpha \in \mathbb{Q}$. We conjecture, in analogy with the Riemann zeta function, that $M_k(T;\alpha )\sim c_k(\alpha )T(\log T{)}^{k^2}$ . Using heuristics from analytic number theory and random matrix theory, we conjecturally compute $c_k(\alpha )$. In the process, we investigate moments of products of Dirichlet $L$-functions on the critical line. We prove our conjectures for the cases $k=1,2$.

arXiv comment37 pages, incorporates referee's comments. (To appear in Math. Proc. Cambridge Philos. Soc.)