{
  "id": "1404.4007",
  "version": "v3",
  "published": "2014-04-15T18:22:16.000Z",
  "updated": "2014-05-05T14:18:24.000Z",
  "title": "Bounded gaps between primes with a given primitive root",
  "authors": [
    "Paul Pollack"
  ],
  "comment": "small corrections to the treatment of \\sum_1 on pp. 11--12",
  "categories": [
    "math.NT"
  ],
  "abstract": "Fix an integer $g \\neq -1$ that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which $g$ is a primitive root. Forty years later, Hooley showed that Artin's conjecture follows from the Generalized Riemann Hypothesis (GRH). We inject Hooley's analysis into the Maynard--Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer $m \\geq 2$. If $q_1 < q_2 < q_3 < \\dots$ is the sequence of primes possessing $g$ as a primitive root, then $\\liminf_{n\\to\\infty} (q_{n+(m-1)}-q_n) \\leq C_m$, where $C_m$ is a finite constant that depends on $m$ but not on $g$. We also show that the primes $q_n, q_{n+1}, \\dots, q_{n+m-1}$ in this result may be taken to be consecutive.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-05-05T14:18:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11N05"
    ],
    "keywords": [
      "primitive root",
      "bounded gaps",
      "inject hooleys analysis",
      "finite constant",
      "perfect square"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.4007P"
    }
  }
}