{
  "id": "math/0401223",
  "version": "v5",
  "published": "2004-01-18T17:24:00.000Z",
  "updated": "2008-11-06T18:44:11.000Z",
  "title": "The distribution of integers with a divisor in a given interval",
  "authors": [
    "Kevin Ford"
  ],
  "comment": "Final version. Greatly simplified proof of Lemma 4.7 in Sec. 10, references updated, other minor corrections",
  "journal": "Annals of Math. (2) 168 (2008), 367-433",
  "categories": [
    "math.NT"
  ],
  "abstract": "We determine the order of magnitude of H(x,y,z), the number of integers n\\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\\le x^{0.49}. For every r\\ge 2, $C>1$ and $\\epsilon>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\\log y)^{\\log 4 -1 - \\epsilon} \\le z \\le \\min(y^{C},x^{1/2-\\epsilon}). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and several related conjectures. One key element of the proofs is a new result on the distribution of uniform order statistics.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2008-11-06T18:44:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N25",
      "62G30"
    ],
    "keywords": [
      "distribution",
      "uniform order statistics",
      "related conjectures",
      "consequence"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1223F"
    }
  }
}