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 pi )/2 pi ln 2 (where x=(E/2 pi )(ln(E/2 pi )-1)+7/8), V is the variance of the Gaussian unitary ensemble (GUE) of random matrices, while when L>>Lmax, 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 Em near x=1012 confirm all details of the semiclassical predictions to within the limits of graphical precision.