Is there quantum chaos in the prime numbers?

TL;DR

This paper investigates whether prime numbers exhibit signs of quantum chaos by analyzing their statistical properties, revealing a transition from chaotic to regular behavior as the number of primes increases.

Contribution

It provides the first detailed statistical analysis of prime numbers indicating a possible connection to quantum chaos and discusses the physical interpretation of primes as quantum eigenvalues.

Findings

Statistical measures show a transition from random matrix to Poisson statistics with increasing N.

Number variance saturates at large lengths, consistent with eigenvalue sequences.

Prime numbers may be modeled as eigenvalues of a quantum system with chaotic and regular dynamics.

Abstract

A statistical analysis of the prime numbers indicates possible traces of quantum chaos. We have computed the nearest neighbor spacing distribution, number variance, skewness, and excess for sequences of the first N primes for various values of N. All four statistical measures clearly show a transition from random matrix statistics at small N toward Poisson statistics at large N. In addition, the number variance saturates at large lengths as is common for eigenvalue sequences. This data can be given a physical interpretation if the primes are thought of as eigenvalues of a quantum system whose classical dynamics is chaotic at low energy but regular at high energy. We discuss some difficulties with this interpretation in an attempt to clarify what kind of physical system might have the primes as its quantum eigenvalues.