Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl 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 t n have a similar structure to those of the semiclassical En ; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n 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 the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.

Abstract Aspects of quantum chaos are studied experimentally, utilising analogies between acoustic systems and quantum systems. These analogies have their origin in the wave nature of sound propagation inside a finite, solid resonator and are clearly expressed through the striking similarities observed for the spectral statistics of acoustic resonators and quantum systems. Such similarities can be understood within the framework of Random Matrix Theory (RMT), which predicts universality for the fluctuation properties of eigenvalue spectra of chaotic systems. The present study includes highly significant tests of well known predictions from Random Matrix Theory for the spectral statistics. The obtained data are in every respect of higher quality than in previous studies of acoustic systems and of higher statistical significance than other experimental studies in the same field. Very recently, studies of parametric level motion have attracted a lot of interest and several analytical results have been obtained within the framework of Random Matrix Theory. This presentation will include studies of quantities related to level velocity and curvature. In particular, we present for the first time a highly significant experimental study of the velocity auto correlation function. Several RMT predictions exist for the statistical properties of wave functions of a quantum particle trapped inside an infinite well of "chaotic " geometry. As a model system we consider standing waves in elastic plates and present a unique experimental technique for measuring the amplitude of such waves. The results presented here show for the first time that the analogies between quantum systems and acoustic systems are not limited to results for the eigenvalue spectra; they include also wave function properties. Finally, we present the distributions of amplitudes and widths for acoustic resonance spectra. For the amplitudes, the result is compared to the Porter-Thomas distribution. For the widths, we compare the results to a simple prediction, suggested by a RMT model.

We measure ultrasound transmission spectra and standing wave patterns of homogeneous and isotropic plates with free boundaries. Our work takes two directions. First, we study plates with the shape of known chaotic billiards, where we focus on issues of universality for spectral uctuations and level motion. We also study the effect of breaking the center-plane symmetry of the plate on spectral statistics and resonance widths, comparing our results to a random matrix model for approximate symmetries. This part of our work may be termed 'acoustic chaology', relating directly to issues studied in the field of quantum chaos. Second, we study a rectangular plate, where we focus on properties like mode conversion, acoustic bouncing-ball states, that are particular to these freely vibrating plates, but of general importance for elastodynamic systems. We seek to understand properties of these systems through a statistical approach and this part of our work may be termed 'statistical elastodynamics'...