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 the deep literature on moments of the Riemann zeta function, before talking about the case of Hurwitz with 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 established that the fourth moment is 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 $3$).
Zoom link: https://icts-res-in.zoom.us/j/93231938657?pwd=Jl8GbklJTX7totWStWhCJMDKBLxYyn.1
Meeting ID: 932 3193 8657
Passcode: 070708