{
  "id": "0707.0299",
  "version": "v1",
  "published": "2007-07-02T20:08:26.000Z",
  "updated": "2007-07-02T20:08:26.000Z",
  "title": "The distribution of smooth numbers in arithmetic progressions",
  "authors": [
    "K. Soundararajan"
  ],
  "comment": "15 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a wide range of $x$ and $y$ we show that ${\\Cal S}(x,y)$, the set of integers below $x$ composed only of prime factors below $y$, is equidistributed in the reduced residue classes $\\pmod q$ for all $q<y^{4\\sqrt{e}-\\epsilon}$. This improves earlier work of Granville; any improvement of this range of $q$ would have interesting consequences for Vinogradov's conjecture on the least quadratic non-residue. For larger ranges of $q$ we prove the existence of a large subgroup of the group of reduced residues such that ${\\Cal S}(x,y)$ is equidistributed within cosets of that subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-07-02T20:08:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic progressions",
      "smooth numbers",
      "distribution",
      "wide range",
      "prime factors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.0299S"
    }
  }
}