{
  "abstract": "For \\(n\\ge2\\), let \\(A(n)=π(n)-ω(n)\\) and \\(B(n)=ϕ(n)-π(n)+ω(n)\\) denote the numbers of prime and nonprime totatives of \\(n\\), respectively, with \\(1\\) included in \\(B(n)\\). We construct an explicit injectively parametrized family \\(\\mathcal C_n\\) of composite totatives by completing suitably restricted squarefree products of prime totatives with a larger prime totative that is the unique largest prime factor, making the parametrization recoverable. We prove \\(\\lvert\\mathcal C_n\\rvert/A(n)\\ge (e^{-γ}-o(1))\\log A(n)/\\log\\log A(n)\\), realizing the sharp classical leading constant \\(e^{-γ}\\). For the same family, with the same defining parameters, we obtain an effective refinement with error of order \\((\\log\\log A(n))^{-1/4}\\) and absolute effectively computable constants. Classical minimal-order results and the prime number theorem show that \\(e^{-γ}\\) is optimal: no family contained in the nonprime totatives can satisfy a uniform lower bound at this scale with a larger leading constant. We also determine exact eventual thresholds. Let \\(N_k\\) and \\(M_k\\) be the least integers such that \\(ϕ(n)>kπ(n)\\) for every \\(n\\ge N_k\\) and \\(B(n)>kA(n)\\) for every \\(n\\ge M_k\\), respectively. We prove \\(M_k\\le N_{k+1}\\) for every \\(k\\ge1\\), determine \\(N_1,\\ldots,N_6\\) and \\(M_1,\\ldots,M_5\\) exactly, and obtain \\(M_k=N_{k+1}\\) for \\(1\\le k\\le5\\).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13852",
  "author_total": 1,
  "authors": [
    {
      "name": "Amirali Fatehizadeh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/94r7mn8tazdrm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13852",
  "fulltext_url": null,
  "id": "94r7mn8tazdrm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13852",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.13852",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/94r7mn8tazdrm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13852",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13852"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Prime and Nonprime Totatives: A Sharp Construction and Exact Thresholds",
  "title_text": "Prime and Nonprime Totatives: A Sharp Construction and Exact Thresholds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/94r7mn8tazdrm",
  "venue": "arXiv",
  "versions": []
}