{
  "id": "2505.01645",
  "version": "v1",
  "published": "2025-05-03T01:19:00.000Z",
  "updated": "2025-05-03T01:19:00.000Z",
  "title": "Note on a sum involving the divisor function",
  "authors": [
    "Liuying Wu"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. In this paper, we prove that $$\\sum_{n\\leq x^{1/c}}d\\left(\\left[\\frac{x}{n^c}\\right]\\right)=d_cx^{1/c}+\\mathcal{O}_{\\varepsilon,c} \\left(x^{\\max\\{(2c+2)/(2c^2+5c+2),5/(5c+6)\\}+\\varepsilon}\\right),$$ where $d_c=\\sum_{k\\geq1}d(k)\\left(\\frac{1}{k^{1/c}}-\\frac{1}{(k+1)^{1/c}}\\right)$ is a constant. This result constitutes an improvement upon that of Feng.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-03T01:19:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divisor function",
      "integral part",
      "real number",
      "result constitutes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}