HP Labs Technical Reports

The Riemann Zeros and Eigenvalue Asymptotics

Berry, M.V.; Keating , J.P.

HPL-BRIMS-98-26

Keyword(s): Riemann hypothesis; semiclassical asymptotics

Abstract: Please Note. This abstract contains mathematical formulae which cannot be represented here. Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eignevalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl . Many features of Hcl are provided by the analogy; for example, the 'Riemann dynamics' should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and universal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to he trajectories generated by the classical hamiltonian Hcl =XP. Notes: M.V.Berry, H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL UK.

65 Pages

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