Semiclassical formula for the number variance of the Riemann zeros

By pretending that the imaginary 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 = ln(E/2 pi)/2 pi In 2 (where x = (E/2 pi)(ln(E/2 pi) 1) + 7/8), Vis the variance of the Gaussian unitary ensemble (GUE) of random matrices, while when L >> L-max, V will have quasirandom oscillations about the mean value pi(-2)(ln In(E/2 pi) + 1.4009). Comparisons with V(L; x) computed by Odlyzko from 10(5) zeros E-m near x = 10(12)' confirm all details of the semiclassical predictions to within the limits of graphical precision.