{
  "id": "1103.2106",
  "version": "v1",
  "published": "2011-03-10T19:03:52.000Z",
  "updated": "2011-03-10T19:03:52.000Z",
  "title": "On a paper of K. Soundararajan on smooth numbers in arithmetic progressions",
  "authors": [
    "Adam J. Harper"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In a recent paper, K. Soundararajan showed, roughly speaking, that the integers smaller than x whose prime factors are less than y are asymptotically equidistributed in arithmetic progressions to modulus q, provided that y^{4\\sqrt{e}-\\delta} \\geq q and that y is neither too large nor too small compared with x. We show that these latter restrictions on y are unnecessary, thereby proving a conjecture of Soundararajan. Our argument uses a simple majorant principle for trigonometric sums to handle a saddle point that is close to 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-10T19:03:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic progressions",
      "smooth numbers",
      "soundararajan",
      "simple majorant principle",
      "saddle point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.2106H"
    }
  }
}