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.

On the critical points of random matrix characteristic polynomials and of the Riemann ξ-function

“…Here we also mention the works devoted to the zeros of ζ , particularly, [47,48,15,43] (and references therein). The zeros of ζ do not lie on the critical line, and are therefore thematically more distant from the current study; their counterparts in random matrix setting (in a sense made precise in the aforementioned works) are the critical points of the characteristic polynomial of a circular ensemble.…”

Landau–Siegel zeros and zeros of the derivative of the Riemann zeta function

“…The conclusion of the theorem is weaker than m(ν) > 0 for ν > 0, but only by a factor of log (2) T . Thus, it is more than sufficient to apply the results of Conrey and Iwaniec [1].…”

Lehmer Pairs Revisited

“…Following the methods of [5], we will develop an argument to support our expectation that when (γ + − γ − ) log(γ 0 )/(2π), the normalized gap, is small (notation as above), then…”