Hidden Multiscale Order in the Primes

Abstract

We study the {pair correlations between} prime numbers in an interval $M\le p\le M+L$ with $M\to \infty$, $L/M\to \beta >0$. 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.

arXiv comment12 figures

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