Contents
** Next:** 1. INTRODUCTION

# A nonlinear equation and its application to nearest neighbor
spacings for zeros of the zeta function and eigenvalues of random matrices

**P.J. Forrester**

Department of
Mathematics

University of Melbourne

Parkville, Victoria 3052

Australia

**
A.M. Odlyzko
**

AT&T Bell Laboratories

Murray Hill

New Jersey 07974

U.S.A.

### Abstract:

*A nonlinear equation generalizing the form of the Painlevé V
equation is used
to compute the probability density function for the distance from an eigenvalue
of a matrix from the GUE ensemble to the eigenvalue nearest to it. (The
classical results concern distribution of the distances between consecutive
eigenvalues.) Comparisons are made with the corresponding distribution for zeros
of the Riemann zeta function, which are conjectured to behave like
eigenvalues of large random GUE matrices.
*

omp@cecm.sfu.ca