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Applications of statistical mechanics in number theory

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  • Wolf, Marek

Abstract

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.

Suggested Citation

  • Wolf, Marek, 1999. "Applications of statistical mechanics in number theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 274(1), pages 149-157.
    • Handle: RePEc:eee:phsmap:v:274:y:1999:i:1:p:149-157
    DOI: 10.1016/S0378-4371(99)00318-0
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    1. Gadiyar, H.Gopalkrishna & Padma, R., 1999. "Ramanujan–Fourier series, the Wiener–Khintchine formula and the distribution of prime pairs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 269(2), pages 503-510.
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    Cited by:

    1. Cattani, Carlo & Ciancio, Armando, 2016. "On the fractal distribution of primes and prime-indexed primes by the binary image analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 222-229.
    2. Vito Barbarani, 2021. "Combinatorial Models of the Distribution of Prime Numbers," Mathematics, MDPI, vol. 9(11), pages 1-50, May.

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