Moments of the Hurwitz zeta function on the critical line

TL;DR

This paper investigates the moments of the Hurwitz zeta function on the critical line, conjectures their asymptotic behavior, and proves the conjectures for the first two moments, linking number theory and random matrix theory.

Contribution

It introduces conjectures for the moments of the Hurwitz zeta function and proves them for the cases k=1,2, advancing understanding of their asymptotic behavior.

Findings

Conjectured asymptotic formula for moments of Hurwitz zeta function.

Proved the conjectures for the first two moments.

Connected moments of Hurwitz zeta with moments of Dirichlet L-functions.

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 $k = 1,2$.