Mathematics Research Centre, Queen Mary Department of Mathematics, Brunel University

"RANDOM MATRICES AND RELATED TOPICS"

Monthly Colloquia

Date and time: 11 October 2002, 16:30

Location: Room G2, School of Mathematical Science, Queen Mary, University of London

Speaker: Dr. F. Mezzadri (Bristol)

"Random matrix theory and the zeros of d\zeta(s)/ds"

Abstract:

The behaviour of the non-trivial zeros of the Riemann zeta function is closely related to the properties of the zeros of its derivative. Indeed, a theorem by Speiser (1934) states that the Riemann hypothesis is equivalent to the statement that $d\zeta(s)/ds$ does not vanish to the left of the critical line. Furthermore, work by Montgomery and Levinson (1974), and Conrey et al (1989) has shown that the distribution of the zeros of $d\zeta(s)/ds$ significantly affects the ability to determine the fraction of zeros of the zeta function on the critical line. However, very little is known about such distribution.

Following previous random matrix theory models of the Riemann zeta function, in this talk we assume the the density of the zeros of $d \zeta(s) /ds$ can be modelled by the roots of the derivative of the characteristic polynomial Z(U,z) of a random unitary matrix with distribution given by Haar measure on the unitary group. We show that as the dimension of the matrix tends to infinity, the fraction of roots of dZ(U,z)/dz that lie in the region 1 - x/(N-1)< |z| < 1 tends to a limit function and derive asymptotic expressions for this function in the limits x -> 0 and x ->infinity.