Applications of statistical mechanics in number theory

The links between statistical physics and number theory are discussed. First the attempts to prove the Riemann Hypothesis by means of the suitable spin model and the Lee–Yang theorem about zeros of the partition function are shortly reviewed. Next, the analogies between random walks and prime numbers are mentioned. In the last section the partition function of the system whose energies are defined by the distances between consecutive primes is calculated. The arguments are given that such a “prime numbers gas” behaves like a set of noninteracting harmonic oscillators.