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.

• A list of mathactivations

• 1. INTRODUCTION
• 2. A NON-LINEAR EQUATION
  • 2.1 Summary of results
  • 2.2 Derivation
  • 2.3 Theory of Tracy and Widom
• 3. STATISTICS OF THE ZEROS OF THE RIEMANN ZETA FUNCTION
  • 3.1 Solution of the non-linear equation
  • 3.2 Empirical value of for Riemann zeta function zeros
• Table 1
• Figure captions
• References
• About this document ...