Phase Space Distribution of Riemann Zeros

TL;DR

This paper derives a phase space distribution for the zeros of the Riemann zeta function by relating eigenvalues of unitary matrices to free fermion phase space, revealing a universal topological feature of matrix models.

Contribution

It introduces a novel phase space framework connecting Riemann zeros to unitary matrix models and free fermion distributions, providing new insights into their structure.

Findings

Phase space distribution for Riemann zeros derived

Universal topological features of unitary matrix phases identified

Relation between eigenvalues and Young tableaux established

Abstract

We present the partition function of a most generic $U(N)$ single plaquette model in terms of representations of unitary group. Extremising the partition function in large N limit we obtain a relation between eigenvalues of unitary matrices and number of boxes in the most dominant Young tableaux distribution. Since, eigenvalues of unitary matrices behave like coordinates of free fermions whereas, number of boxes in a row is like conjugate momenta of the same, a relation between them allows us to provide a phase space distribution for different phases of the unitary model under consideration. This proves a universal feature that all the phases of a generic unitary matrix model can be described in terms of topology of free fermi phase space distribution. Finally, using this result and analytic properties of resolvent that satisfy Dyson-Schwinger equation, we present a phase space distribution of unfolded zeros of Riemann zeta function.