{
  "abstract": "For a natural number $n>1$ let $P^-(n)$ denote its smallest prime divisor. Alladi and Johnson proved that $\\sum_{P^{-}(n)\\equiv l \\Mod{k}} \\frac{μ(n)ω(n)}{n} = 0$ for $(l,k)=1$, and Tenenbaum showed that the partial sums of this series up to $X$ are $O_\\varepsilon\\bigl((\\log\\log X)^{5/3+\\varepsilon}/\\log X\\bigr)$. In this article, we prove that these partial sums are in fact equal to $C(k,l)/\\log X + O\\bigl((\\log\\log X)^4/(\\log X)^2\\bigr)$, with an explicit constant $C(k,l)$. We also investigate the behaviour of these constants. In particular, we show under GRH that the number of reduced classes $l\\Mod{k}$ for which $C(k,l)<0$ is $φ(k)(1- o(1))$ as $k\\to\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23660",
  "author_total": 1,
  "authors": [
    {
      "name": "Deep Thakur"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eeyveq3qan3r8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23660",
  "fulltext_url": null,
  "id": "eeyveq3qan3r8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23660",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7222979027",
  "pdf_url": "https://arxiv.org/pdf/2610.23660",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eeyveq3qan3r8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23660",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23660"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a series related to the Möbius function",
  "title_text": "On a series related to the Möbius function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/eeyveq3qan3r8",
  "venue": "arXiv",
  "versions": []
}