{
  "id": "2107.08308",
  "version": "v1",
  "published": "2021-07-17T19:48:08.000Z",
  "updated": "2021-07-17T19:48:08.000Z",
  "title": "Reciprocity Relations for Summations of Squares of Floor Functions and Fractional Parts of Fractions",
  "authors": [
    "Damanvir Singh Binner"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given positive coprime integers $a$ and $b$ and a natural number $h$, we obtain reciprocity relations which can be used to quickly evaluate summations like $\\sum_{i=1}^{h} \\{\\frac{ib}{a}\\}^2$ and $\\sum_{i=1}^{h} \\lfloor \\frac{ib}{a} \\rfloor^2$, where $\\lfloor x \\rfloor$ and $\\{x\\}$ denote the floor function and the fractional part of $x$, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-17T19:48:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional part",
      "floor function",
      "reciprocity relations",
      "quickly evaluate summations",
      "positive coprime integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}