Correlations of eigenvalues and Riemann zeros

TL;DR

This paper introduces a new method based on the ratios conjecture to explicitly compute lower order terms in the correlation of Riemann zeta zeros and eigenvalues of random matrices, unifying their structures.

Contribution

It provides a novel approach to derive explicit formulas for lower order correlation terms assuming the ratios conjecture, applicable to both zeta zeros and random matrix eigenvalues.

Findings

Derived explicit formulas for lower order correlation terms.

Unified the structure of correlations between zeta zeros and random matrix eigenvalues.

Validated the approach for both number theory and random matrix theory contexts.

Abstract

We present a new approach to obtaining the lower order terms for $n$-correlation of the zeros of the Riemann zeta function. Our approach is based on the `ratios conjecture' of Conrey, Farmer, and Zirnbauer. Assuming the ratios conjecture we prove a formula which explicitly gives all of the lower order terms in any order correlation. Our method works equally well for random matrix theory and gives a new expression, which is structurally the same as that for the zeta function, for the $n$-correlation of eigenvalues of matrices from U(N).