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The Riemann zeros and eigenvalue asymptotics. (English) Zbl 0928.11036

The purpose of this paper is to report on the development of an analogy involving three different areas of mathematics and physics, namely eigenvalue asymptotics in wave (and quantum) physics, dynamical chaos, and prime number theory.

At the heart of the matter is a longstanding speculation concerning the non-trivial zeros of the Riemann zeta-function, which are related to the vibration frequencies (or quantum energies) of some wave system, underlying which there is a dynamical system whose rays (or trajectories) are chaotic.

A precise identification of this dynamical system would lead directly to a proof of the Riemann hypothesis. As of now, the system is yet unknown, but many of its properties have already been discovered, yielding insights both from mathematics to physics – by stimulating the development of new spectral asymptotics – and from physics to mathematics - by indicating unsuspected correlations between the Riemann zeros.

These connections have been reviewed before.

In the paper under review, however, they include several recent developments concerning, in particular, the statistics of the zeros and quantum eigenvalues (developed in section 4 of the paper), the description of a powerful method for calculating the heights ${t}_{n}$ of the Riemann zeros (the Riemann-Siegel formula), what is done in section 5 and involves a physical interpretation in terms of resurgence of long periodic orbits – that implies new interpretations of the periodic-orbit sum for quantum spectra – and a list of the properties of the conjectured dynamical system (given in section 6 of the paper), with a final speculation that the Riemann zeros are eigenvalues of some quantization of the classical dynamical system generated by the Hamiltonian ${H}_{cl}=XP$.

##### MSC:
 11M26 Nonreal zeros of $\zeta \left(s\right)$ and $L\left(s,\chi \right)$; Riemann and other hypotheses 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 35P20 Asymptotic distribution of eigenvalues and eigenfunctions for PD operators 81Q50 Quantum chaos 81U99 Scattering theory (quantum theory) 37K99 Infinite-dimensional Hamiltonian systems