{
  "id": "math/0111012",
  "version": "v1",
  "published": "2001-11-01T21:54:48.000Z",
  "updated": "2001-11-01T21:54:48.000Z",
  "title": "Spacing of zeros of Hecke L-functions and the class number problem",
  "authors": [
    "J. Brian Conrey",
    "Henryk Iwaniec"
  ],
  "comment": "61 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We derive strong and effective lower bounds for the class number h(q) of the imaginary quadratic field Q(\\sqrt{-q}), conditionally subject to the existence of many small (subnormal) gaps between zeros of the L-function associated with a character of the class group associated with this field. In particular, we prove that if the gap between consecutive zeros of the L-function is somewhat smaller than the average for sufficiently many pairs of zeros on the critical line, then h >> \\sqrt q (log q)^{-A} for some constant A > 0. For the trivial character, the L-function is the Dedekind zeta-function of the number field and so contains the Riemann zeta-function as a factor. Thus, as a corollary to our main result, we prove that this lower bound for h follows from the hypothesis that there are sufficiently many pairs of adjacent zeros of the Riemann zeta-function on the critical line whose spacing is slightly smaller than 1/2 of the average spacing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-11-01T21:54:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "class number problem",
      "hecke l-functions",
      "riemann zeta-function",
      "imaginary quadratic field",
      "critical line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math.....11012C"
    }
  }
}