On moments of the derivative of CUE characteristic polynomials and the Riemann zeta function

Abstract

We study the derivative of the characteristic polynomial of $N \times N$ Haar distributed unitary matrices. We obtain the first explicit formulae for complex-valued moments when the spectral variable is inside the unit disc, in the limit $N \to \infty$. These formulae are expressed in terms of the confluent hypergeometric function of the first kind. As an application, we provide an alternative method to re-obtain Mezzadri's result [J. Phys. A, 36(12):2945-2962, 2003] on the asymptotic density of zeros of the derivative as $N \to \infty$. We explore the connection between these moments and those of the derivative of the Riemann zeta function away from the critical line. Under the Lindel\"of hypothesis, we prove that all positive integer moments agree with our random matrix results up to an arithmetic factor. Inspired by this finding, we propose a conjecture on the asymptotics of non-integer moments of the derivative of the Riemann zeta function off the critical line. Within random matrix theory, we also investigate the microscopic regime where the spectral variable $z$ satisfies $|z|^{2}=1-\frac{c}{N}$ for a fixed constant $c$. We obtain an asymptotic formula for the moments in this regime as a determinant involving the finite temperature Bessel kernel, which reduces to the Bessel kernel when $c=0$. For finite matrix size, we provide an exact formula for the moments of the derivative inside the unit disc, expressed as polynomials of the inverse of the distance from the circle, with coefficients given by combinatorial sums.