Moments of the Hurwitz zeta function on the critical line

Abstract

We study the moments M k (T ; α) = 2T T |ζ(s, α)| 2k dt of the Hurwitz zeta function ζ(s, α) on the critical line, s = 1/2+it.We conjecture, in analogy with the Riemann zeta function, that M k (T ; α) ∼ c k (α)T (log T ) k 2 . In the case of α ∈ Q, we use heuristics from analytic number theory and random matrix theory to compute c k (α). 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 α ∈ Q.

A paucity problem associated with a shifted integer analogue of the divisor function

“…For irrational shifts θ, this forms the focus of an ongoing project of the first and second author. For rational shifts θ, see [3].…”