Correlations between large prime numbers

TL;DR

This paper investigates the statistical correlations among large prime numbers, revealing a Poissonian nature at very large scales and suggesting a chaotic hypothesis for moderate primes, with correlation length increasing logarithmically.

Contribution

It demonstrates the transition from non-Poissonian to Poissonian correlations in prime numbers and proposes a new chaotic hypothesis for moderate primes.

Findings

Correlation length  4.5 for primes ^5

Correlation length increases logarithmically with prime size

Poissonian distribution applies only to very large primes

Abstract

It is shown that short-range correlations between large prime numbers (~10^5 and larger) have a Poissonian nature. Correlation length \zeta = 4.5 for the primes ~10^5 and it is increasing logarithmically according to the prime number theorem. For moderate prime numbers (~10^4) the Poissonian distribution is not applicable while the correlation length surprisingly continues to follow to the logarithmical law. A chaotic (deterministic) hypothesis has been suggested to explain the moderate prime numbers apparent randomness.