Mathematics Colloquium

The prime numbers have been a source of fascination for millennia and continue to surprise us. Motivated by the fact that nontrivial zeros of the Riemann zeta function constitutes a disordered and hyperuniform point process, 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 "effectively limit-periodic" [1,2]. The primes
in this regime are hyperuniform but are ordered in the sense described above. This is shown analytically using the structure factor S(k),proportional to the scattering intensity from a many-particle system, where k is the wavenumber. Remarkably, the structure factor for primes is characterized by dense Bragg peaks (Dirac delta functions), 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. I will begin with a short review of the 
hyperuniformity concept and then discuss the main results concerning the primes.

1. S. Torquato, G. Zhang, and M. de Courcy-Ireland, "Uncovering Multiscale Order in the Prime Numbers via Scattering, Journal of Statistical Mechanics: Theory and Experiment, 2018, 093401 (2018).

2. S. Torquato, G. Zhang, and M. de Courcy-Ireland, "Hidden Multiscale Order in the Primes", arXiv:1804.06279.