{
  "id": "2106.13983",
  "version": "v1",
  "published": "2021-06-26T09:35:43.000Z",
  "updated": "2021-06-26T09:35:43.000Z",
  "title": "A generalization of Tóth identity in the ring of algebraic integers involving a Dirichlet Character",
  "authors": [
    "Subha Sarkar"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The $k$-dimensional generalized Euler function $\\varphi_k(n)$ is defined to be the number of ordered $k$-tuples $(a_1,a_2,\\ldots, a_k) \\in \\mathbb{N}^k$ with $1\\leq a_1,a_2,\\ldots, a_k \\leq n$ such that both the product $a_1a_2\\cdots a_k$ and the sum $a_1+a_2+\\cdots+a_k$ are co-prime to $n$. T\\'oth proved that the identity \\begin{equation*} \\sum_{\\substack{a_1,a_2,\\ldots, a_k=1 \\\\ \\gcd(a_1a_2\\cdots a_k,n)=1\\\\ \\gcd(a_1+a_2+\\cdots+a_k,n)=1}}^n \\gcd(a_1+a_2+\\cdots+a_k-1,n) =\\varphi_k(n)\\sigma_0(n), \\;\\; \\text{ where } \\sigma_s(n) = \\sum_{d\\mid n}d^s \\;\\; \\text{ holds. } \\end{equation*} This identity can also be viewed as a generalization of Menon's identity. In this article, we generalize this identity to the ring of algebraic integers involving arithmetical functions and Dirichlet characters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-26T09:35:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11A25"
    ],
    "keywords": [
      "algebraic integers",
      "dirichlet character",
      "tóth identity",
      "generalization",
      "dimensional generalized euler function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}