The fourth moment of the Hurwitz zeta function

TL;DR

This paper establishes a sharp upper bound for the fourth moment of the Hurwitz zeta function on the critical line with irrational shift parameter, revealing growth behavior similar to Gaussian distribution conjectures.

Contribution

It provides the first sharp bounds for the fourth moment of the Hurwitz zeta function with irrational shifts and determines the order of magnitude for moments up to the 4th.

Findings

The fourth moment grows like T (log T)^k for 0 ≤ k ≤ 2.

The growth behavior differs from the Riemann zeta function, indicating a Gaussian distribution conjecture.

The results apply when the shift parameter has irrationality exponent less than 3.

Abstract

We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line when the shift parameter $\alpha$ is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the $2k$th moment for all $0 \leqslant k \leqslant 2$ in this case. In contrast to the Riemann zeta function and other $L$-functions from arithmetic, these grow like $T (\log T)^k$. This suggests, and we conjecture, that the value distribution of $\zeta(s,\alpha)$ on the critical line is Gaussian.