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

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

2010-02-02

arxiv.org/abs/1002.0372v1

Nonlinearity, Vol. 23 (2010), 2599-2621

Mathematics

Number Theory

24 pages, 6 figures

Scientific paper

10.1088/0951-7715/23/10/014

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.

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