Nearest-neighbor-spacing distribution of prime numbers and quantum chaos
Abstract
We give heuristic arguments and computer results to support the hypothesis that, after appropriate rescaling, the statistics of spacings between adjacent prime numbers follows the Poisson distribution. The scaling transformation removes the oscillations in the nearest-neighbor-spacing distribution of primes. These oscillations have the very profound period of length six. We also calculate the spectral rigidity Δ3 for prime numbers by two methods. After suitable averaging one of these methods gives the Poisson dependence Δ3(L)=L/15.
- Publication:
-
Physical Review E
- Pub Date:
- February 2014
- DOI:
- arXiv:
- arXiv:1212.3841
- Bibcode:
- 2014PhRvE..89b2922W
- Keywords:
-
- 05.45.Mt;
- 02.10.De;
- Quantum chaos;
- semiclassical methods;
- Algebraic structures and number theory;
- Mathematics - Number Theory;
- Mathematical Physics
- E-Print:
- Many changes incorporated: Gallagher theorem mentioned, in Sect. IV the averaging over "probabilistic'' primes and Fig.8 are added (p. 8). The spectral rigidity averaged over 100 realizations of these artificial primes displays perfect $L/15$ dependence. 11 Figures