The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, with a shift parameter \(0 < \alpha \leqslant 1\). We will consider moments of the Hurwitz zeta function on the critical line with a focus on the case where the shift \(\alpha\) is irrational. We will briefly review rational \(\alpha\), which leads naturally into moments of products of Dirichlet \(L\)-functions. Heuristics involving random matrix theory can then be used to predict an asymptotic formula for all integer moments.

For irrational \(\alpha\), we will discuss recent work joint with Winston Heap investigating these moments, where we proved a sharp upper bound for the fourth moment of the order \(T(\log T)^2\) assuming that \(\alpha\) is not too well-approximable by rationals (concretely, when its irrationality exponent \(\mu(\alpha)\) is less than \(3\)).

We also put forth a conjecture for higher moments that suggests that the distribution of the Hurwitz zeta function with irrational shifts on the critical line is approximately Gaussian. This contrasts with the Riemann zeta function (and other \(L\)-functions from arithmetic), where the analogous fourth moment is of order \(T(\log T)^4\) and where the distribution is approximately log-Gaussian instead of Gaussian.