Matrix Model for Riemann Zeta via its Local Factors

TL;DR

This paper constructs a novel matrix model for the Riemann zeta function by assembling prime-specific unitary matrices, linking number theory with quantum Hamiltonians and phase space analysis.

Contribution

It introduces a prime-by-prime construction of a matrix ensemble for the zeta function, integrating $p$-adic phase space and Berry-Keating type Hamiltonians.

Findings

Proposes a $p$-adic phase space approach to zeta function modeling.

Constructs a Hamiltonian inspired by Berry-Keating conjecture.

Combines prime-specific data into a unified matrix model.

Abstract

We propose the construction of an ensemble of unitary random matrices (UMM) for the Riemann zeta function. Our approach to this problem is `$p$-iecemeal', in the sense that we consider each factor in the Euler product representation of the zeta function to first construct a UMM for each prime $p$. We are able to use its phase space description to write the partition function as the trace of an operator that acts on a subspace of square-integrable functions on the $p$-adic field. This suggests a Berry-Keating type Hamiltonian. We combine the data from all primes to propose a Hamiltonian and a matrix model for the Riemann zeta function.