{
  "id": "1804.06279",
  "version": "v1",
  "published": "2018-04-16T00:49:02.000Z",
  "updated": "2018-04-16T00:49:02.000Z",
  "title": "Hidden Multiscale Order in the Primes",
  "authors": [
    "Salvatore Torquato",
    "Ge Zhang",
    "Matthew De Courcy-Ireland"
  ],
  "comment": "12 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the {pair correlations between} prime numbers in an interval $M \\leq p \\leq M + L$ with $M \\rightarrow \\infty$, $L/M \\rightarrow \\beta > 0$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-16T00:49:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hidden multiscale order",
      "structure factor",
      "dyadic interval",
      "implies anomalously suppressed density fluctuations",
      "dirac delta functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}