{
  "id": "1709.04799",
  "version": "v1",
  "published": "2017-09-14T14:00:45.000Z",
  "updated": "2017-09-14T14:00:45.000Z",
  "title": "The maximal order of iterated multiplicative functions",
  "authors": [
    "Christian Elsholtz",
    "Marc Technau",
    "Niclas Technau"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Following Wigert, a great number of authors including Ramanujan, Gronwall, Erd\\H{o}s, Ivi\\'{c}, Heppner, J. Knopfmacher, Nicolas, Schwarz, Wirsing, Freiman, Shiu et al. determined the maximal order of several multiplicative functions, generalizing Wigert's result \\[\\max_{n\\leq x} \\log d(n)= (\\log 2+o(1))\\frac{\\log x}{\\log \\log x}.\\] On the contrary, for many multiplicative functions, the maximal order of iterations of the functions remains wide open. The case of the iterated divisor function was only recently solved, answering a question of Ramanujan (1915). Here, we determine the maximal order of $\\log f(f(n))$ for a class of multiplicative functions $f$ which are related to the divisor function. As a corollary, we apply this to the function counting representations as sums of two squares of non-negative integers, also known as $r_2(n)/4$, and obtain an asymptotic formula: \\[\\max_{n\\leq x} \\log f(f(n))= (c+o(1))\\frac{\\sqrt{\\log x}}{\\log \\log x},\\] with some explicitly given positive constant $c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-14T14:00:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37"
    ],
    "keywords": [
      "maximal order",
      "iterated multiplicative functions",
      "functions remains wide open",
      "great number",
      "generalizing wigerts result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}