Gaussian unitary ensemble eigenvalues and Riemann [Formula Presented] function zeros: A nonlinear equation for a new statistic

Abstract

For infinite Gaussian unitary ensemble random matrices the probability density function [Formula Presented] for the nearest neighbor eignenvalue spacing (as distinct from the spacing between consecutive eigenvalues) is computed in terms of the solution of a certain nonlinear equation, which generalizes the σ form of the Painlevé [Formula Presented] equation. Comparison is made with the empirical value of [Formula Presented] for the zeros of the Riemann [Formula Presented] function on the critical line, including data from [Formula Presented] consecutive zeros near zero number [Formula Presented]. © 1996 The American Physical Society.