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

TL;DR

This paper explores the unexpected bimodal distribution of zeros of the derivatives of the Riemann zeta function and characteristic polynomials, linking number theory and random matrix theory with new proofs and conjecture support.

Contribution

It proves Mezzadri's conjecture on the leading order behavior of zeros of characteristic polynomial derivatives and connects this to the zeta function zeros, revealing a common bimodal distribution.

Findings

Bimodal distribution observed in both cases.

Proof of Mezzadri's conjecture for unitary matrices.

Connection established between zeta zeros and random matrix conjectures.

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.