Uncovering Multiscale Order in the Prime Numbers via Scattering

Abstract

The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call {\it effectively limit-periodic}. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor $S(k)$, proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor for primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. We identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes.

arXiv comment5 figures

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Cited by in corpus (11)

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• Hyperuniformity Order Metric of Barlow Packings
• Effective delocalization in the one-dimensional Anderson model with stealthy disorder
• Hyperuniformity in Ashkin-Teller model
• Computing Riemann zeros with light scattering
• Hydrodynamic stability and pattern formation in hexatic epithelial layers
• Diffraction of the primes and other sets of zero density
• Precise Determination of the Long-Time Asymptotics of the Diffusion Spreadability of Two-Phase Media