GAUSSIAN UNITARY ENSEMBLE EIGENVALUES AND RIEMANN ZETA-FUNCTION ZEROS- A NONLINEAR EQUATION FOR A NEW STATISTIC

For infinite Gaussian unitary ensemble random matrices the probability density function S-nn(t) for the nearest neighbor eignenvalue spacing (as distinct from the spacing between consecutive eigenvalues) is com puted in terms of the solution of a certain nonlinear equation, which generalizes the sigma form of the Painleve V equation. Comparison is m ade with the empirical value of S-nn(t) for the zeros of the Riemann z eta function on the critical line, including data from 10(6) consecuti ve zeros near zero number 10(20).