{
  "id": "1805.04940",
  "version": "v1",
  "published": "2018-05-13T20:13:50.000Z",
  "updated": "2018-05-13T20:13:50.000Z",
  "title": "Negative moments of the gaps between consecutive primes",
  "authors": [
    "Marek Wolf"
  ],
  "comment": "Apparently for the first time the closed formula for the sum of reciprocals of gaps between consecutive primes is presented",
  "categories": [
    "math.NT"
  ],
  "abstract": "We derive heuristically formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x $ represented directly by $\\pi(x)$ --- the number of primes up to $x$. In particular we propose a closed formula for the sum of reciprocals of gaps between consecutive primes $<x : ~ M_{-1}(x)\\sim \\frac{\\pi^2(x)}{x-2\\pi(x)}\\log\\Big(\\frac{x}{2\\pi(x)}\\Big) \\sim x \\log \\log(x)/\\log^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\\times 10^{18}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-13T20:13:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A41",
      "11N05",
      "11L20"
    ],
    "keywords": [
      "consecutive primes",
      "negative moments",
      "enormous computer data",
      "derive heuristically formulas",
      "reciprocals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}