{
  "id": "1311.4600",
  "version": "v3",
  "published": "2013-11-19T01:05:10.000Z",
  "updated": "2019-10-28T23:08:00.000Z",
  "title": "Small gaps between primes",
  "authors": [
    "James Maynard"
  ],
  "comment": "25 pages; corrected typos",
  "categories": [
    "math.NT"
  ],
  "abstract": "We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method, and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, $\\liminf_{n}(p_{n+m}-p_n)<\\infty$ for any integer $m$. We also show that $\\liminf(p_{n+1}-p_n)\\le 600$, and, if we assume the Elliott-Halberstam conjecture, that $\\liminf_n(p_{n+1}-p_n)\\le 12$ and $\\liminf_n (p_{n+2}-p_n)\\le 600$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-20T17:37:12.000Z",
      "comment": "23 pages; Mathematica file included with source documents; Error on page 21 fixed",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2019-10-28T23:08:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N35",
      "11N36"
    ],
    "keywords": [
      "small gaps",
      "gpy sieve method",
      "tuples conjecture holds",
      "elliott-halberstam conjecture",
      "refinement avoids"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.4600M"
    }
  }
}