{
  "id": "1706.04028",
  "version": "v1",
  "published": "2017-06-13T12:38:52.000Z",
  "updated": "2017-06-13T12:38:52.000Z",
  "title": "The variance of the Euler totient function",
  "authors": [
    "Tom van Overbeeke"
  ],
  "comment": "15 pages, 3 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we study the variance of the Euler totient function (normalized to $\\varphi(n)/n$) in the integers $\\mathbb{Z}$ and in the polynomial ring $\\mathbb{F}_q[T]$ over a finite field $\\mathbb{F}_q$. It turns out that in $\\mathbb{Z}$, under some assumptions, the variance of the normalized Euler function becomes constant. This is supported by several numerical simulations. Surprisingly, in $\\mathbb{F}_q[T]$, $q\\rightarrow \\infty$, the analogue does not hold: due to a high amount of cancellation, the variance becomes inversely proportional to the size of the interval.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-13T12:38:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T55",
      "11M38",
      "11M50",
      "11N37"
    ],
    "keywords": [
      "euler totient function",
      "finite field",
      "normalized euler function",
      "polynomial",
      "assumptions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}