Semiclassical formula for the number variance of the Riemann zeros

Abstract

By pretending that the imaginery parts E_m 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<>L_max, V will have quasirandom oscillations about the mean value pi ^-2(lnln(E/2 pi )+1.4009). Comparisons with V(L;x) computed by Odlyzko (1987) from 10⁵ zeros E_m near x=10¹² confirm all details of the semiclassical predictions to within the limits of graphical precision.