{
  "id": "2008.05837",
  "version": "v1",
  "published": "2020-08-13T11:48:39.000Z",
  "updated": "2020-08-13T11:48:39.000Z",
  "title": "A disproof of Hooley's conjecture",
  "authors": [
    "Daniel Fiorilli",
    "Greg Martin"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Define $G(x;q)$ to be the variance of primes $p\\le x$ in the arithmetic progressions modulo $q$, weighted by $\\log p$. Hooley conjectured that as soon as $q$ tends to infinity and $x\\ge q$, we have the upper bound $G(x;q) \\ll x \\log q$. In this paper we show that the upper bound does not hold in general, and that $G(x;q)$ can be asymptotically as large as $x (\\log q+\\log\\log\\log x)^2/4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-13T11:48:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N13",
      "11M26"
    ],
    "keywords": [
      "hooleys conjecture",
      "upper bound",
      "arithmetic progressions modulo"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}