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

Abstract We study the derivative of the characteristic polynomial of Haar‐distributed unitary matrices. We obtain new explicit formulae for complex‐valued moments when the spectral variable is inside the unit disc, in the limit . These formulae are expressed in terms of the confluent hypergeometric function of the first kind. We explore the connection between these moments and those of the derivative of the Riemann zeta function away from the critical line. Under the Lindelöf hypothesis, we prove that all positive integer moments agree with our random matrix results up to a well‐known arithmetic factor. Inspired by this finding, we propose a conjecture on the asymptotics of noninteger 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 satisfies for a fixed constant . 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 .