It seems that the distribution of eigenvalues in quantum chaotic systems obeys the same statistics as eigenvalues of random matrices. e.g.
http://www.researchgate.net/publication/51969774_Generalized_random_matrix_conjecture_for_chaotic_systems
It has also been shown that the distribution of the critical zeros of the Riemann zeta-function can be related to the distribution of eigenvalues of certain random matrices:
http://www.researchgate.net/publication/51940315_Random_matrices_and_Riemann_hypothesis
So this would seem to to suggest a statistical relationship between the zeros of the zeta function and the energy levels some quantum chaotical system. Can any relationship be drawn between the placement of the prime numbers and a dynamical system? Any other implications of this connection?
Article Generalized random matrix conjecture for chaotic systems
Article Random matrices and Riemann hypothesis
The person who thinks about these sorts of questions is Michel Planat: http://hal.archives-ouvertes.fr/hal-00123212/
The following attachment (based on a seminar given at Cerfim in 2007) might be of some interest to you. Prof. Sekatskii has collaborated with Prof. D. Merlini (Cerfim) who is also working on these questions.
The Euler totient function which counts the relative prime numbers with respect to given number appears in the description of large N gauge theories in the free limit. See http://arxiv.org/abs/hep-th/0310285 and http://arxiv.org/abs/hep-th/0305052
At the beginning of this paper there is a brief review of different areas of physics where primes numbers come into play: http://arxiv.org/pdf/cond-mat/0303110
Perhaps, if the most fundamental theory will be ever formulated, it should be so fundamental that it should contain no dimensional or dimensionless constant. Then its properties will entirely rely on the number theory, in particular the complexity of prime number distribution.
I am surprized that everybody has forgotten the old work of Alain Connes and Benoit Bost.
http://www.alainconnes.org/docs/bostconnesscan.pdf
on which my work with Planat and Omar is based
http://iopscience.iop.org/1751-8121/labtalk-article/45421
Of the BC-system has not been realized in the laboratory yet. It is **theoretical **physics.
I believe that proof of RH connects pure Mathematics to the pure & applied physics, but do not have any clue showing off its application. I think that if one can find a relation, it shall make a revolution in physics and science.
Berry and Keating have conjectured that there should exist a chaotic system such that the periods of the periodic orbits are the logarithms of the prime numbers. The quantization of this system would have the Riemann zeroes as eigtenvalues.
I think about the probability theory. Nevertheless M. Planck, A. Einstein and E. Schrödinger were right! They were fully developed people. And here M. Born and L. Landau aren't right because at them (at Born and Landau) the right cerebral hemisphere in a cherpny box was underdeveloped. So the theory and experiment showed.
The universe is causal, so there are no "random" numbers or matrices.
Albeit, the distribution of the prime numbers IS related to a physical system, by the Schrödinger equation.
Quantum mechanics defines the rules, and otherwise "pure" mathematics has to follow. You can see more in my profile at RG.
Here is an approach on that subject.
[PDF] Prime Numbers: A Particle in a Box and the Complex Wave Model | Semantic Scholar
Prime Numbers: A Particle in a Box and the Complex Wave Model - Archive ouverte HAL https://hal.science/hal-01230473v1
Preprint The Universal Fractal Chirality of Riemann Zeta-Function: On...
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