Matrix models for zeta functions

Riemann hypothesized that the zeta function Z(s) has (non-trivial)
zeroes only on the line Re(s)=1/2. Hilbert and Polya suggested that
the position of these zeroes might be related to the spectrum of a
`Hamiltonian’. It has been known for some time that the statistical
properties of the eigenvalue distribution of an ensemble of random
matrices resemble those of the zeroes of the zeta function. We
construct a unitary matrix model (UMM) for the zeta function,
however, our approach to the problem is “piecemeal”. That is, we
consider each factor in the Euler product representation of the zeta
function to get a UMM for each prime. This suggests a Hamiltonian (of
the type proposed by Berry-Keating and Connes) from its phase space
description. We attempt to combine this to get a matrix model for the
full zeta function. This talk will be based on work in progress with
A. Chattopadhyay, P. Dutta and S. Dutta.