The structure factor of primes

TL;DR

This paper investigates the spatial correlations of prime numbers using structure factor analysis, revealing ordered, quasicrystal-like patterns and providing explicit formulas for peak locations, challenging previous notions of randomness.

Contribution

It introduces a numerical analysis of prime pair statistics using structure factors, uncovering ordered patterns and explicit peak formulas, and compares primes to quasicrystals.

Findings

Primes exhibit Bragg-like peaks in their structure factor.

The peaks' locations and heights follow an explicit formula.

The diffuse background decays slowly, vanishing in the infinite limit.

Abstract

Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes using statistical-mechanical methods, especially the structure factor $S(k)$ in an interval $M \leq p \leq M + L$ with $M$ large, and $L/M$ smaller than unity. We show that the structure factor of the prime-number configurations in such intervals exhibits well-defined Bragg-like peaks along with a small "diffuse" contribution. This indicates that the primes are appreciably more correlated and ordered than previously thought. Our numerical results definitively suggest an explicit formula for the locations and heights of the peaks. This formula predicts infinitely many peaks in any non-zero interval, similar to the behavior of quasicrystals. However, primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational. We also show numerically that the diffuse part decays slowly as $M$ and $L$ increases. This suggests that the diffuse part vanishes in an appropriate infinite-system-size limit.