{
  "id": "2406.04018",
  "version": "v1",
  "published": "2024-06-06T12:44:03.000Z",
  "updated": "2024-06-06T12:44:03.000Z",
  "title": "Inequalities involving the primorial counting function",
  "authors": [
    "Christian Axler"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\varphi(n)$ denote the Euler totient function. In this paper, we first establish a new upper bound for $n/\\varphi(n)$ involving $K(n)$, the function that counts the number of primorials not exceeding $n$. In particular, this leads to an answer to a question raised by Aoudjit, Berkane, and Dusart concerning an upper bound for the sum-of-divisors function $\\sigma(n)$. Furthermore, we give some lower bounds for $N_k/\\varphi(N_k)$ as well as for $\\sigma(N_k)/N_k$, where $N_k$ denotes the $k$th primorial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-06T12:44:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "primorial counting function",
      "inequalities",
      "upper bound",
      "euler totient function",
      "th primorial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}