Gaussian unitary ensemble eigenvalues and Riemannζfunction zeros: A nonlinear equation for a new statistic

For infinite Gaussian unitary ensemble random matrices the probability density function $S_{\mathrm{nn}}\mathrm{}\left(t\right)\mathrm{}$ 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 $V$ equation. Comparison is made with the empirical value of $S_{\mathrm{nn}}\mathrm{}\left(t\right)\mathrm{}$ for the zeros of the Riemann $\zeta$ function on the critical line, including data from ${10}^6$ consecutive zeros near zero number ${10}^{20}$.