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

TL;DR

This paper conjectures the behavior of moments of mixed derivatives of the Riemann zeta function at its zeros, using two methods: characteristic polynomials in CUE and a hybrid model approach, providing consistent asymptotic predictions.

Contribution

It introduces conjectures for the moments of derivatives of the zeta function at zeros, derived via random matrix theory and hybrid models, unifying two different analytical approaches.

Findings

Asymptotic formulas for moments of characteristic polynomial derivatives in CUE.

Consistent conjectures from both random matrix and hybrid model approaches.

Insights into the behavior of derivatives of the Riemann zeta function at zeros.

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.