Nathan Creighton

Mixed moments of the Riemann zeta function

Moments of the Riemann zeta function play a central role in understanding the horizontal distribution of its zeroes, while mixed moments involving its derivatives shed light on the gaps between consecutive zeroes. Motivated by the random matrix model for the zeta function, Assiotis, Gunes, Keating and Wei (2026) established asymptotic formulae for general joint moments of characteristic polynomials and their higher-order derivatives, a problem recently revisited by Snape using combinatorial methods. In this talk, I investigate the corresponding joint moments of the Riemann zeta function and its higher derivatives. I show that two distinct approaches—the recipe of Conrey et al. and the method involving moments of long Dirichlet polynomials—lead to the same conjectural asymptotics.