Moments of derivatives of the Riemann zeta function: Characteristic polynomials and the hybrid formula

Abstract

We conjecture results about the moments of mixed derivatives of the Riemann zeta function, evaluated at the non-trivial zeros of the Riemann zeta function. We do this in two different ways, both giving us the same conjecture. In the first, we find asymptotics for the moments of derivatives of the characteristic polynomials of matrices in the Circular Unitary Ensemble. In the second, we consider the hybrid model approach first proposed by Gonek, Hughes and Keating.

The work in this paper has been superseded by two new papers. One method of proof has been removed, and the remaining results are split between Complex Moments of the Derivative of the Riemann zeta function (arXiv:2509.07788) and Integer Moments of the Derivatives of the Riemann zeta function (arXiv:2509.07792). These also include alternative methods justifying the original conjectures/RMT