{
  "id": "2401.13118",
  "version": "v1",
  "published": "2024-01-23T21:57:47.000Z",
  "updated": "2024-01-23T21:57:47.000Z",
  "title": "The first and second moment for the length of the period of the continued fraction expansion for $\\sqrt{d}$",
  "authors": [
    "Francesco Battistoni",
    "Loïc Grenié",
    "Giuseppe Molteni"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $d$ be any positive non square integer. We prove an upper bound for the first two moments of the length $T(d)$ of the period of the continued fraction expansion for $\\sqrt{d}$. This allows to improve the existing results for the large deviations of $T(d)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-23T21:57:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A55",
      "11R11",
      "11Y65"
    ],
    "keywords": [
      "continued fraction expansion",
      "second moment",
      "positive non square integer",
      "upper bound",
      "large deviations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}