Semiclassical formula for the number variance of the Riemann zeros
- 1 August 1988
- journal article
- Published by IOP Publishing in Nonlinearity
- Vol. 1 (3) , 399-407
- https://doi.org/10.1088/0951-7715/1/3/001
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, 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.Keywords
This publication has 6 references indexed in Scilit:
- The Bakerian Lecture, 1987. Quantum chaologyProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1987
- On the distribution of spacings between zeros of the zeta functionMathematics of Computation, 1987
- Semiclassical theory of spectral rigidityProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1985
- Periodic orbits and a correlation function for the semiclassical density of statesJournal of Physics A: General Physics, 1984
- Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillationsAnnals of Physics, 1972
- Periodic Orbits and Classical Quantization ConditionsJournal of Mathematical Physics, 1971