2018/01/04 by Ge Zhang, Fausto Martelli, Salvatore Torquato
Materials Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Combinatorics #Condensed matter physics #Geometry #Infinity #Interval (graph theory) #Limit (mathematics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Physics #Prime (order theory) #Prime number #Quasicrystal #Quasicrystal Structures and Properties #Statistical physics #Structure factor #Zero (linguistics) #cond-mat.stat-mech
paper · pdf · doi:10.1088/1751-8121/aaa52a
published as J. Stat. Phys. A 51, 11 (2018)
openalex publication_date 2018/01/04 · arxiv created 2018/02/14 · arxiv updated 2018/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes using statistical–mechanical methods, particularly the structure factor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>S</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> in an interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>M</mml:mi> <mml:mo>⩽</mml:mo> <mml:mi>p</mml:mi> <mml:mo>⩽</mml:mo> <mml:mi>M</mml:mi> <mml:mo>+</mml:mo> <mml:mi>L</mml:mi> </mml:mstyle> </mml:math> with M large, and L / M smaller than unity. We show that the structure factor of the prime-number configurations in such intervals exhibits well-defined Bragg-like peaks along with a small ‘diffuse’ contribution. This indicates that primes are appreciably more correlated and ordered than previously thought. Our numerical results definitively suggest an explicit formula for the locations and heights of the peaks. This formula predicts infinitely many peaks in any non-zero interval, similar to the behavior of quasicrystals. However, primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational. We also show numerically that the diffuse part decays slowly as M and L increases. This suggests that the diffuse part vanishes in an appropriate infinite-system-size limit.