{
  "id": "1803.03377",
  "version": "v1",
  "published": "2018-03-09T04:14:53.000Z",
  "updated": "2018-03-09T04:14:53.000Z",
  "title": "On the sum of the reciprocals of the differences between consecutive primes",
  "authors": [
    "Nian Hong Zhou"
  ],
  "comment": "5 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p_n$ denote the $n$-th prime number, and let $d_n=p_{n+1}-p_{n}$. Under the Hardy--Littlewood prime-pair conjecture, we prove \\begin{align*} \\sum_{n\\le X}\\frac{\\log^{\\alpha}d_n}{d_n}\\sim\\begin{cases} \\quad\\frac{X\\log\\log\\log X}{\\log X}~\\qquad\\quad~ &\\alpha=-1, \\frac{X}{\\log X}\\frac{(\\log\\log X)^{1+\\alpha}}{1+\\alpha}\\qquad &\\alpha>-1, \\end{cases} \\end{align*} and establish asymptotic properties for some series of $d_n$ without the Hardy--Littlewood prime-pair conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-09T04:14:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N36",
      "11A41"
    ],
    "keywords": [
      "consecutive primes",
      "hardy-littlewood prime-pair conjecture",
      "differences",
      "reciprocals",
      "th prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}