The fourth moment of the Hurwitz zeta function

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.