The Riemann zeros and eigenvalue asymptotics

Comparison between formulae for the counting functions of the heights t(n) of the Riemann zeros and of semiclassical quantum. eigenvalues E-n suggests that the t(n) are eigenvalues of an (unknown) hermitean operator H, obtain ed by quantizing a, classical dynamical system with hamiltonian H-cl. Many features of H-cl are provided by the analogy; for example, the "Riemann dyn amics" should be chaotic and have periodic orbits whose periods are multipl es of logarithms of prime numbers. Statistics of the t(n) have a similar st ructure to those of the semiclassical E-n; in particular, they display rand om-matrix universality at short range, and nonuniversal behaviour over long er ranges. Very refined features of the statistics of the t(n) can be compu ted accurately from formulae with quantum analogues. The Riemann-Siegel for mula 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 the trajectories generated by the classical hamiltonian H-cl = XP.