Correlations of eigenvalues and Riemann zeros

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).