On the logarithmic derivative of characteristic polynomials for random unitary matrices

Abstract

Let $U\in U(N)$ be a random unitary matrix of size $N$, distributed with respect to the Haar measure on $U(N)$. Let $P(z)=P_U(z)$ be the characteristic polynomial of $U$. We prove that for $z$ close to the unit circle, $\frac{P^{\prime }}{P}(z)$ can be approximated using zeros of $P$ very close to $z$, 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 $\frac{P^{\prime }}{P}(z)$ away from the unit circle, and this is an analogue of a result of Lester for zeta.

arXiv commentfixed an inaccuracy in the proof of Prop. 2.1

References in corpus (5)

• Mixed moments of characteristic polynomials of random unitary matrices
• A central limit theorem for the zeroes of the zeta function
• Bootstrapped zero density estimates and a central limit theorem for the zeros of the zeta function
• Moments of the logarithmic derivative of characteristic polynomials from $SO(N)$ and $USp(2N)$
• On the logarithmic derivative of characteristic polynomials for random unitary matrices

Cited by in corpus (1)

• On the logarithmic derivative of characteristic polynomials for random unitary matrices