{
  "id": "1510.04577",
  "version": "v1",
  "published": "2015-10-15T15:19:07.000Z",
  "updated": "2015-10-15T15:19:07.000Z",
  "title": "A note on the distribution of normalized prime gaps",
  "authors": [
    "János Pintz"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let us denote the nth difference between consecutive primes by d_n. The Prime Number Theorem clearly implies that d_n is logn on average. Paul Erd\\H{o}s conjectured about 60 years ago that the sequence d_n/logn is everywhere dense on the nonnegative part of the real line. He and independently G. Ricci proved in 1954-55 that the set J of limit points of the sequence {d_n/logn} has positive Lebesgue measure. The first and until now only concrete known element of J was proved to be the number zero in the work of Goldston, Yildirim and the present author. The author of the present note showed in 2013 (arXiv: 1305.6289) that there is a fixed interval containing 0 such that all elements of it are limit points. In 2014 it was shown by W.D. Banks, T. Freiberg and J. Maynard (arXiv: 1404.5094) that one can combine the Erd\\H{o}s-Rankin method (producing large prime gaps) and the Maynard-Tao method (producing bounded prime gaps) to obtain the lower bound (1+o(1))T/8 for the Lebesgue measure of the subset of limit points not exceeding T. In the present note we improve this lower bound to (1+o(1))T/4, using a refinement of the argument of Banks, Freiberg and Maynard.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-15T15:19:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "normalized prime gaps",
      "limit points",
      "prime number theorem clearly implies",
      "lower bound",
      "distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151004577P"
    }
  }
}