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Matrix Models for Zeta Functions

By mluzum - Posted on 20 abril 2018

Prof. Debashis Ghoshal (School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India)
Segunda-feira, 30 Julho, 2018 - 11:00

Riemann hypothesized that the zeta function $\zeta(s)$ has (non-trivial) zeroes only on the line $\mathrm{Re}(s)=\frac{1}{2}$ in the complex $s$-plane. 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 models (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 and Keating) from its phase space description. We attempt to combine this to get a matrix model for the full zeta function.