{
  "id": "1504.05549",
  "version": "v1",
  "published": "2015-04-21T19:05:08.000Z",
  "updated": "2015-04-21T19:05:08.000Z",
  "title": "Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method",
  "authors": [
    "Sary Drappeau"
  ],
  "comment": "44 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Building on classical work of Deshouillers and Iwaniec and recent work of Blomer and Mili\\'cevi\\'c, we prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the \"smooth\" summation variables. We employ this estimate to obtain power-saving in the dispersion method, in the setting of Bombieri, Fouvry, Friedlander and Iwaniec. As a consequence, assuming the Riemann hypothesis for Dirichlet $L$-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating $\\sum_{p\\leq x}\\tau(p-1)$. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and Tenenbaum, we obtain power-saving in the asymptotic formula for $\\sum_{n\\leq x}\\tau_k(n)\\tau(n+1)$, reproving a result announced by Bykovski\\u{i} and Vinogradov by a different method. The gain in the exponent is shown to be independent of $k$ if a generalized Lindel\\\"of hypothesis is assumed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-21T19:05:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kloosterman sums",
      "dispersion method",
      "arithmetic progressions",
      "titchmarsh divisor problem",
      "congruence conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150405549D"
    }
  }
}