*(English)*Zbl 1167.81017

From the text: A brief review of recent developments in the theory of the Riemann zeta-function inspired by ideas and methods of quantum chaos is given.

At the first glance the Riemann zeta-function and quantum chaos are completely disjoint fields. The Riemann zeta-function is a part of pure number theory but quantum chaos is a branch of theoretical physics devoted to the investigation of non-integrable quantum problems like the hydrogen atom in external fields.

Nevertheless for a long time it was understood that there exist multiple interrelations between these two subjects. In Sections 2 and 3 the Riemann and the Selberg zeta-functions and their trace formulae are informally compared. From the comparison it appears that in many aspects zeros of the Riemann zeta function resemble eigenvalues of an unknown quantum chaotic Hamiltonian.

One of the principal tools in quantum chaos is the investigation of statistical properties of deterministic energy levels of a given Hamiltonian. In such approach one stresses not precise values of physical quantities but their statistics by considering them as different realizations of a statistical ensemble. According to the BGS conjecture energy levels of chaotic quantum systems have the same statistical properties as eigenvalues of standard random matrix ensembles depended only on the exact symmetries. In Section 4 it is argued that is quite natural to conjecture that statistical properties of the Riemann zeros are the same as of eigenvalues of the Gaussian unitary ensemble of random matrices (GUE). This conjecture is very well confirmed by numerics but only partial rigorous results are available.

In Section 5 a semiclassical method which permits, in principle, to calculate correlation functions is shortly discussed. The main problem here is to control correlations between periodic orbits with almost the same lengths.

In Sections 6 and 7 it is demonstrated how the Hardy-Littlewood conjecture about distribution of near-by primes leads to explicit formula for the two-point correlation function of the Riemann zeros. The resulting formula describes non-universal approach to the GUE result in excellent agreement with numerical results.

In Section 8 it is demonstrated how to calculate non-universal corrections to the nearest-neighbor distribution for the Riemann zeros.

Spectral statistics is not the only interesting statistical characteristics of zeta functions. The mean moments of the Riemann zeta-function along the critical line is another important subject that attracts wide attention in number theory during a long time.

In Section 9 it is explained how random matrix theory permit Keating and Snaith to propose the breakthrough conjecture about mean moments. This conjecture now is widely accepted and is generalized for different zeta and $L$-functions and different quantities as well.

##### MSC:

81Q50 | Quantum chaos |

11M50 | Relations with random matrices |

11M06 | $\zeta \left(s\right)$ and $L(s,\chi )$ |

81-02 | Research monographs (quantum theory) |