{
  "id": "1301.5663",
  "version": "v2",
  "published": "2013-01-23T22:32:07.000Z",
  "updated": "2013-04-16T15:21:10.000Z",
  "title": "The distribution of the variance of primes in arithmetic progressions",
  "authors": [
    "Daniel Fiorilli"
  ],
  "comment": "26 pages; Modified Definition 2.1, the error term for the variance in Theorem 1.2 and its proof",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "abstract": "Hooley conjectured that the variance V(x;q) of the distribution of primes up to x in the arithmetic progressions modulo q is asymptotically x log q, in some unspecified range of q\\leq x. On average over 1\\leq q \\leq Q, this conjecture is known unconditionally in the range x/(log x)^A \\leq Q \\leq x; this last range can be improved to x^{\\frac 12+\\epsilon} \\leq Q \\leq x under the Generalized Riemann Hypothesis (GRH). We argue that Hooley's conjecture should hold down to (loglog x)^{1+o(1)} \\leq q \\leq x for all values of q, and that this range is best possible. We show under GRH and a linear independence hypothesis on the zeros of Dirichlet L-functions that for moderate values of q, \\phi(q)e^{-y}V(e^y;q) has the same distribution as that of a certain random variable of mean asymptotically \\phi(q) log q and of variance asymptotically 2\\phi(q)(log q)^2. Our estimates on the large deviations of this random variable allow us to predict the range of validity of Hooley's Conjecture.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-04-16T15:21:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N13",
      "60F10",
      "11M26"
    ],
    "keywords": [
      "distribution",
      "hooleys conjecture",
      "linear independence hypothesis",
      "arithmetic progressions modulo",
      "large deviations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.5663F"
    }
  }
}