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

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.

18 pages, 5 figures. Revised version: section 4 expanded; minor corrections

References in corpus (3)

• Random matrix theory and discrete moments of the Riemann zeta function
• Saddle points in the chaotic analytic function and Ginibre characteristic polynomial
• Linear statistics for zeros of Riemann's zeta function

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• Moments of the logarithmic derivative of characteristic polynomials from $SO(N)$ and $USp(2N)$
• Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices
• A new approach to the characteristic polynomial of a random unitary matrix
• Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function
• Developments in Random Matrix Theory
• Joint moments of derivatives of characteristic polynomials
• Moments of the derivative of the Riemann zeta-function and of characteristic polynomials