Nearest neighbor spacing distribution of prime numbers and quantum chaos

TL;DR

This paper investigates the statistical distribution of spacings between prime numbers, providing heuristic and computational evidence that, after rescaling, their distribution follows a Poisson pattern similar to quantum chaos phenomena.

Contribution

It introduces a rescaling method to analyze prime gaps, demonstrating their spacing distribution aligns with Poisson statistics and exploring spectral rigidity in primes.

Findings

Prime spacings follow Poisson distribution after rescaling.

Oscillations in prime spacings have a period of six.

Spectral rigidity $3$ approximates L/15 after averaging.

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 NNSD of primes. These oscillations have the very profound period of length six. We also calculate the spectral rigidity $\Delta_3$ for prime numbers by two methods. After suitable averaging one of these methods gives the Poisson dependence $\Delta_3(L)=L/15$.