Roots of the derivative of the Riemann zeta function and of characteristic polynomials

Abstract

We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function.

24 pages, 6 figures

References in corpus (3)

• How to generate random matrices from the classical compact groups
• Random matrix theory and the zeros of zeta'(s)
• Derivatives of random matrix characteristic polynomials with applications to elliptic curves

Cited by in corpus (11)

• A matrix model with a singular weight and Painleve' III
• Mixed moments of characteristic polynomials of random unitary matrices
• Zeros of random polynomials and its higher derivatives
• On the critical points of random matrix characteristic polynomials and of the Riemann $\xi$-function
• Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices
• On an inequality for the Riemann zeta-function in the critical strip
• On the distribution of the zeros of the derivative of the Riemann zeta-function
• Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function
• Lehmer pairs revisited
• X-Ray of Zhang's eta function
• Simple Zeros Of The Zeta Function