On the logarithmic derivative of characteristic polynomials for random unitary matrices

Abstract

AbstractLet be a random unitary matrix of size , distributed with respect to the Haar measure on . Let be the characteristic polynomial of . We prove that for close to the unit circle, can be approximated using zeros of very close to , with a typically controllable error term. This is an analogue of a result of Selberg for the Riemann zeta‐function. We also prove a mesoscopic central limit theorem for away from the unit circle, and this is an analogue of a result of Lester for zeta.