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\nRiemann zeta function and compare this to the radial distribution of zeros of\nthe derivative of the characteristic polynomial of a random unitary matrix.\nBoth cases show a surprising bimodal distribution which has yet to be\nexplained. We show by example that the bimodality is a general phenomenon. For\nthe unitary matrix case we prove a conjecture of Mezzadri concerning the\nleading order behavior, and we show that the same follows from the random\nmatrix conjectures for the zeros of the zeta function.\n