The fourth moment of the Hurwitz zeta function

We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function (s, ) on the critical line when the shift parameter is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the 2kth 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 (T) ᵏ. This suggests, and we conjecture, that the value distribution of (s, ) on the critical line is Gaussian.