Riemann zeros and quantum chaos

This article will describe properties of Riemann zeros and their links with field of quantum chaos. The article is reserved for the invited expert.

Short note and Related References added by Scholarpedia Editor D.Shepelyansky in May 2020

Hilbert and Polya put forward the idea that the zeros of the Riemann zeta function may have a spectral origin: the values of \\(t_n\\) such that \\(1/2 +it_n\\) is a non trivial zero of \\(\zeta\\) might be the eigenvalues of a self-adjoint operator. This would imply the Riemann Hypothesis. From the perspective of Physics one might go further and consider the possibility that the operator in question corresponds to the quantization of a classical dynamical system. The significant evidence in support of this spectral interpretation of the Riemann zeros emerged in the 1950’s in the form of the resemblance between the Selberg trace formula, which relates the eigenvalues of the Laplacian and the closed geodesics of a Riemann surface, and the Weil explicit formula in number theory, which relates the Riemann zeros to the primes. More generally, the Weil explicit formula resembles very closely a general class of Trace Formulae, written down by Gutzwiller, that relate quantum energy levels to classical periodic orbits in chaotic Hamiltonian systems. Later the Montgomery calculation of the pair correlation of \\(t_n\\) showed that the zeros exhibit the same repulsion as the eigenvalues of typical large unitary matrices [1], as noted by Dyson. Montgomery conjectured more general analogies with these random matrices, which were confirmed by Odlyzko’s numerical experiments [2]. Later conjectures relating the statistical distribution of random matrix eigenvalues to that of the quantum energy levels of classically chaotic systems connect these two themes. The related research publications are available at [3],[4],[5],[6],[7],[8] (this note is mainly taken from [8]). Recent developments can be found at [9].

## [edit] Related References

1. H.L.Montgomery, "The pair correlation of zeros of the zeta function, Analytic number theory", Proceedings of Symposium in Pure Mathemathics 24 (St. Louis Univ., St.Louis, Mo., 1972), American Mathematical Society (Providence, R.I., 1973), pp.181–193.
2. A.M.Odlyzko, "On the distribution of spacings between the zeros of the zeta function", Math. Comp. 48, 273–308 (1987).
3. M.V.Berry, "Riemann's Zeta function: A model for quantum chaos?", In: T.H.Seligman, H.Nishioka (eds) "Quantum chaos and statistical nuclear physics", Lecture Notes in Physics v.263, Springer, Berlin, Heidelberg.
4. M.V.Berry, "Semiclassical formula for the number variance of the Riemann zeros", Nonlinearity 1: 399 (1988)
5. M.V.Berry, J.P.Keating, "The Riemann zeros and eigenvalue asymptotics", SIAM Rev. 41(2): 236 (1999)
6. E,Bogomolny, "Riemann zeta function and quantum chaos", arXiv:0708.4223 [nlin.CD] (2007)
7. M.Srednicki, "Nonclassical degrees of freedom in the Riemann Hamiltonian", Phys. Rev. Lett. 107: 100201 (2011)
8. P.Bourgade, J.P.Keating, "Quantum Chaos, Random Matrix Theory, and the Riemann \\(\zeta\\)-function", In: B.Duplantier, S.Nonnenmacher, V.Rivasseau (eds), Chaos, Progress in Mathematical Physics, v.66. Birkhauser, Springer, Basel (2013)
9. A.LeClair, "Spectral flow for the Riemann zeros", arXiv:2406.01828 [math-ph] (2024)

## [edit] See also internal links

Bohigas-Giannoni-Schmit conjecture, Microwave billiards and quantum chaos, Quantum chaos, Random matrix theory, Shnirelman theorem