{
  "id": "2105.10844",
  "version": "v1",
  "published": "2021-05-23T01:52:16.000Z",
  "updated": "2021-05-23T01:52:16.000Z",
  "title": "A variant of the prime number theorem",
  "authors": [
    "Kui Liu",
    "Jie Wu",
    "Zhishan Yang"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Lambda(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $\\varepsilon>0$ the asymptotic formula $$ \\sum_{n\\le x} \\Lambda\\Big(\\Big[\\frac{x}{n}\\Big]\\Big) = x\\sum_{d\\ge 1} \\frac{\\Lambda(d)}{d(d+1)} + O_{\\varepsilon}\\big(x^{9/19+\\varepsilon}\\big) \\qquad (x\\to\\infty)$$ holds. This improves a recent result of Bordell\\`es, which requires $\\frac{97}{203}$ in place of $\\frac{9}{19}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-23T01:52:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "prime number theorem",
      "von mangoldt function",
      "asymptotic formula",
      "integral part",
      "real number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}