{
  "id": "1811.03008",
  "version": "v1",
  "published": "2018-11-07T17:03:08.000Z",
  "updated": "2018-11-07T17:03:08.000Z",
  "title": "Limit points of normalized prime gaps",
  "authors": [
    "Jori Merikoski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if $p_n$ denotes the $n$th prime and $\\mathbb{L}$ is the set of limit points of the sequence $\\{(p_{n+1}-p_n)/\\log p_n\\}_{n=1}^\\infty,$ then for all $T\\geq 0$ the Lebesque measure of $\\mathbb{L} \\cap [0,T]$ is at least $T/3.$ This improves the result of Pintz (2015) that the Lebesque measure of $\\mathbb{L} \\cap [0,T]$ is at least $(1/4-o(1))T,$ which was obtained by a refinement of the previous ideas of Banks, Freiberg, and Maynard (2015). Our improvement comes from using Chen's sieve to give, for a certain sum over prime pairs, a better upper bound than what can be obtained using Selberg's sieve. Even though this improvement is small, a modification of the arguments Pintz and Banks, Freiberg, and Maynard shows that this is sufficient. In addition, we show that there exists a constant $C$ such that for all $T \\geq 0$ we have $\\mathbb{L} \\cap [T,T+C] \\neq \\emptyset,$ that is, gaps between limit points are bounded by an absolute constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-07T17:03:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "normalized prime gaps",
      "limit points",
      "lebesque measure",
      "better upper bound",
      "arguments pintz"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}