The fourth moment of the Hurwitz zeta function

We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function ζ(s,α) 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 $2k$th moment for all 0 k 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 ζ(s,α) on the critical line is Gaussian.