{
  "id": "1404.5094",
  "version": "v2",
  "published": "2014-04-21T02:19:45.000Z",
  "updated": "2014-10-20T12:56:19.000Z",
  "title": "On limit points of the sequence of normalized prime gaps",
  "authors": [
    "William D. Banks",
    "Tristan Freiberg",
    "James Maynard"
  ],
  "comment": "Revised and improved",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p_n$ denote the $n$th smallest prime number, and let $\\boldsymbol{L}$ denote the set of limit points of the sequence $\\{(p_{n+1} - p_n)/\\log p_n\\}_{n = 1}^{\\infty}$ of normalized differences between consecutive primes. We show that for $k = 9$ and for any sequence of $k$ nonnegative real numbers $\\beta_1 \\le \\beta_2 \\le ... \\le \\beta_k$, at least one of the numbers $\\beta_j - \\beta_i$ ($1 \\le i < j \\le k$) belongs to $\\boldsymbol{L}$. It follows at least $12.5%$ of all nonnegative real numbers belong to $\\boldsymbol{L}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-21T02:19:45.000Z",
      "abstract": "Let $p_n$ denote the $n$th smallest prime number, and let $\\boldsymbol{L}$ denote the set of limit points of the sequence $ \\left\\{(p_{n+1} - p_n)/\\log p_n\\right\\}_{n = 1}^{\\infty} $ of normalized differences between consecutive primes. We show that for $k = 50$ and for any sequence of $k$ nonnegative real numbers $\\beta_1 \\le \\beta_2 \\le \\cdots \\le \\beta_k$, at least one of the numbers $ \\{\\beta_j - \\beta_i : 1 \\le i < j \\le k\\} $ belongs to $\\boldsymbol{L}$. It follows that more than $2\\%$ of all nonnegative real numbers belong to $\\boldsymbol{L}$.",
      "comment": "19 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-20T12:56:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N36"
    ],
    "keywords": [
      "normalized prime gaps",
      "limit points",
      "th smallest prime number",
      "nonnegative real numbers belong"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.5094B"
    }
  }
}