Hidden Multiscale Order in the Primes

TL;DR

This paper reveals that prime numbers exhibit a multiscale, limit-periodic order and hyperuniformity in their distribution, conditioned on a prime pair conjecture, and introduces an algorithm for prime reconstruction in dyadic intervals.

Contribution

It demonstrates, under a prime pair conjecture, that primes have a multiscale order characterized by a union of periodic systems and develops an algorithm for prime reconstruction.

Findings

Primes show multiscale limit-periodic structure.

Primes exhibit hyperuniformity with suppressed density fluctuations.

An algorithm for high-accuracy prime reconstruction in dyadic intervals is proposed.

Abstract

We study the {pair correlations between} prime numbers in an interval $M \leq p \leq M + L$ with $M \rightarrow \infty$, $L/M \rightarrow \beta > 0$. By analyzing the \emph{structure factor}, we prove, conditionally on the {Hardy-Littlewood conjecture on prime pairs}, that the primes are characterized by unanticipated multiscale order. Specifically, their limiting structure factor is that of a union of an infinite number of periodic systems and is characterized by dense set of Dirac delta functions. Primes in dyadic intervals are the first examples of what we call {\it effectively limit-periodic} point configurations. This behavior implies anomalously suppressed density fluctuations compared to uncorrelated (Poisson) systems at large length scales, which is now known as hyperuniformity. Using a scalar order metric $\tau$ calculated from the structure factor, we identify a transition between the order exhibited when $L$ is comparable to $M$ and the uncorrelated behavior when $L$ is only logarithmic in $M$. Our analysis for the structure factor leads to an algorithm to reconstruct primes in a dyadic interval with high accuracy.