{
  "id": "1511.04468",
  "version": "v1",
  "published": "2015-11-13T21:49:31.000Z",
  "updated": "2015-11-13T21:49:31.000Z",
  "title": "Chains of large gaps between primes",
  "authors": [
    "Kevin Ford",
    "James Maynard",
    "Terence Tao"
  ],
  "comment": "16 pages, no figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p_n$ denote the $n$-th prime, and for any $k \\geq 1$ and sufficiently large $X$, define the quantity $$ G_k(X) := \\max_{p_{n+k} \\leq X} \\min( p_{n+1}-p_n, \\dots, p_{n+k}-p_{n+k-1} ),$$ which measures the occurrence of chains of $k$ consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that \\[ G_1(X) \\gg \\frac{\\log X \\log \\log X\\log\\log\\log\\log X}{\\log \\log \\log X}\\] for sufficiently large $X$. In this note, we combine the arguments in that paper with the Maier matrix method to show that \\[ G_k(X) \\gg \\frac{1}{k^2} \\frac{\\log X \\log \\log X\\log\\log\\log\\log X}{\\log \\log \\log X}\\] for any fixed $k$ and sufficiently large $X$. The implied constant is effective and independent of $k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-13T21:49:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N35"
    ],
    "keywords": [
      "sufficiently large",
      "maier matrix method",
      "consecutive large gaps",
      "th prime",
      "occurrence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151104468F"
    }
  }
}