{
  "id": "1811.02556",
  "version": "v1",
  "published": "2018-11-06T18:57:00.000Z",
  "updated": "2018-11-06T18:57:00.000Z",
  "title": "On error term estimates à la Walfisz for mean values of arithmetic functions",
  "authors": [
    "Yuta Suzuki"
  ],
  "comment": "32 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Walfisz (1963) proved the asymptotic formula \\[ \\sum_{n\\le x}\\varphi(n) = \\frac{3}{\\pi^2}x^2+O(x(\\log x)^{\\frac{2}{3}}(\\log\\log x)^{\\frac{4}{3}}), \\] which improved the error term estimate of Mertens (1874) and had been the best possible estimate for more than 50 years. Recently, H.-Q. Liu (2016) improved Walfisz's error term estimate to \\[ \\sum_{n\\le x}\\varphi(n) = \\frac{3}{\\pi^2}x^2+O(x(\\log x)^{\\frac{2}{3}}(\\log\\log x)^{\\frac{1}{3}}). \\] We generalize Liu's result to a certain class of arithmetic functions and improve the result of Balakrishnan and P\\'etermann (1996). To this end, we provide a refined version of Vinogradov's combinatorial decomposition available for a wider class of multiplicative functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-06T18:57:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N37",
      "11L07"
    ],
    "keywords": [
      "arithmetic functions",
      "mean values",
      "walfiszs error term estimate",
      "vinogradovs combinatorial decomposition",
      "wider class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}