Semiclassical formula for the number variance of the riemann zeros

Abstract

By pretending that the imaginery parts Em of the Riemann zeros are eigenvalues of a quantum Hamiltonian whose corresponding classical trajectories are chaotic and without time-reversal symmetry, it is possible to obtain by asymptotic arguments a formula for the mean square difference V(L;x) between the actual and average number of zeros near the xth zero in an interval where the expected number is L. This predicts that when L<>Lmax=ln(E/2 Π)/2 Π ln 2 (where x=(E/2 Π)(ln(E/2 Π)-1)+7/8), V is the variance of the Gaussian unitary ensemble (GUE) of random matrices, while when L