{
  "id": "1207.5107",
  "version": "v2",
  "published": "2012-07-21T07:11:53.000Z",
  "updated": "2012-10-21T13:55:41.000Z",
  "title": "Evaluation of the convolution sums $\\sum_{l+15m=n} σ(l) σ(m)$ and $\\sum_{3l+5m=n} σ(l) σ(m)$ and some applications",
  "authors": [
    "B. Ramakrishnan",
    "Brundaban Sahu"
  ],
  "comment": "To appear in IJNT",
  "categories": [
    "math.NT"
  ],
  "abstract": "We evaluate the convolution sums $\\sum_{l,m\\in {\\mathbb N}, {l+15m=n}} \\sigma(l) \\sigma(m)$ and $\\sum_{l,m\\in {\\mathbb N}, {3l+5m=n}} \\sigma(l) \\sigma(m)$ for all $n\\in {\\mathbb N}$ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer $n$ by the form $$ x_1^2 + x_1x_2 + x_2^2 + x_3^2 + x_3x_4 + x_4^2 + 5 (x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2). $$ We also determine the number of representations of positive integers by the quadratic form $$ x_1^2 + x_2^2+x_3^2+x_4^2 + 6 (x_5^2+x_6^2+x_7^2+x_8^2), $$ by using the convolution sums obtained earlier by Alaca, Alaca and Williams \\cite{{aw3}, {aw4}}.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-10-21T13:55:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11F11",
      "11E20",
      "11E25",
      "11F20"
    ],
    "keywords": [
      "convolution sums",
      "evaluation",
      "applications",
      "positive integer",
      "quasimodular forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.5107R"
    }
  }
}