Random matrix theory and the zeros of zeta'(s)

TL;DR

This paper investigates the distribution of zeros of the derivative of the characteristic polynomial of random unitary matrices, providing asymptotic results and connections to the zeros of the derivative of the Riemann zeta function.

Contribution

It establishes the limiting distribution of roots of the derivative of characteristic polynomials for large matrices, linking random matrix theory to the zeros of zeta'(s).

Findings

Fraction of roots in a specified region converges as N increases.

Derived asymptotic formulas for the distribution at large and small x.

Numerical experiments support theoretical predictions.

Abstract

We study the density of the roots of the derivative of the characteristic polynomial Z(U,z) of an N x N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function zeta(s), this is expected to be an accurate description for the horizontal distribution of the zeros of zeta'(s) to the right of the critical line. We show that as N --> infinity the fraction of roots of Z'(U,z) that lie in the region 1-x/(N-1) <= |z| < 1 tends to a limit function. We derive asymptotic expressions for this function in the limits x --> infinity and x --> 0 and compare them with numerical experiments.