Hidden multiscale order in the primes

We study the pair correlations between prime numbers in an interval [ image ] with [ image ], [ image ]. By analyzing the 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 a dense set of Dirac delta functions (Bragg peaks), similar to but different from the dense Bragg peaks that arise in quasicrystals and standard limit-periodic systems. Primes in dyadic intervals are the first examples of what we call 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 [ image ] 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 of the structure factor leads to an algorithm to reconstruct primes in a dyadic interval with high accuracy. The discovery of the hyperuniformity and effective limit-periodic behavior of the primes provide new organizing principles to understand the fundamental nature of patterns in the primes.