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

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 determinantal structure of the random matrix limit.