The fourth moment of the Hurwitz zeta function

Abstract

We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function

on the critical line when the shift parameter 𝛼 is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the

2 ⁢ k 2k

th moment for all

0 ⩽ k ⩽ 2 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 T(\log T)^{k}

. This suggests, and we conjecture, that the value distribution of

ζ ⁢ ( s , α ) \zeta(s,\alpha)

on the critical line is Gaussian.