A method for calculating spectral statistics based on random-matrix universality with an application to the three-point correlations of the Riemann zeros

TL;DR

This paper introduces a novel method combining universal random matrix theory and non-universal trace formulas to compute spectral statistics, specifically the three-point correlations of Riemann zeros, supporting heuristic results with a unified approach.

Contribution

It develops a generalized Hermitian random matrix ensemble approach that integrates universal and non-universal spectral contributions for Riemann zeros.

Findings

Derived a three-point correlation function consistent with heuristic methods

Supported the universality of spectral statistics for Riemann zeros

Provided a framework linking random matrix theory with number theory

Abstract

We illustrate a general method for calculating spectral statistics that combines the universal (Random Matrix Theory limit) and the non-universal (trace-formula-related) contributions by giving a heuristic derivation of the three-point correlation function for the zeros of the Riemann zeta function. The main idea is to construct a generalized Hermitian random matrix ensemble whose mean eigenvalue density coincides with a large but finite portion of the actual density of the spectrum or the Riemann zeros. Averaging the random matrix result over remaining oscillatory terms related, in the case of the zeta function, to small primes leads to a formula for the three-point correlation function that is in agreement with results from other heuristic methods. This provides support for these different methods. The advantage of the approach we set out here is that it incorporates the determinental structure of the Random Matrix limit.