{
  "abstract": "We apply the generalized monoid of words and logarithmic-prefix functionals of \\cite{C1} to the regularization of the divergent product \\[ P_ω=\\prod_{n=1}^{\\infty} n^{(-1)^{ω(n)}}, \\] where \\(ω(n)\\) is the number of distinct prime divisors of \\(n\\). The associated Dirichlet series \\(D_ω(s)=\\sum_{n\\ge1}(-1)^{ω(n)}n^{-s}\\) has a natural boundary at \\(\\Re(s)=0\\), as established by Meretov (1987) and recovered here as a special case of the Estermann--Dahlquist theory of Euler products, which makes zeta regularization impossible: the derivative \\(D_ω'(0)\\) does not exist. This product therefore lies completely beyond the reach of classical regularization methods. We construct the canonical net encoding the sign sequence, define the logarithmic-prefix functional with exponent \\(α=2\\), and extract a finite value through the weighted evaluation functional of the monoid framework. The evaluation functional is shown to converge absolutely, and this convergence is unconditional: it follows from the classical zero-free region of \\(ζ(s)\\) (de la Vallée Poussin), which yields the effective bound \\(T_ω(x)=O(x\\,e^{-c\\sqrt{\\log x}})\\) for some \\(c>0\\). Numerical computation up to \\(N=10^7\\) gives the regularized value \\[ P_{ω,\\mathrm{ren}}\\approx 0.7731, \\] a new constant of arithmetic origin that cannot be obtained by any classical summation method. We also compute the second derivative of the Dirichlet series at \\(s=1\\), \\[ D_ω''(1)=8\\,C_2\\ln 2, \\] where \\(C_2\\) is the Hardy--Littlewood twin prime constant, which provides an unconditional analytical link to the arithmetic structure of the sequence \\((-1)^{ω(n)}\\). A self-contained proof of the natural boundary, based on the Estermann--Dahlquist theorem, is given in Appendix~A.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13658",
  "author_total": 2,
  "authors": [
    {
      "name": "A. Alvarez Cruz"
    },
    {
      "name": "E. A. Alvarez Gutierrez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/67ss0q0j5pth5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13658",
  "fulltext_url": null,
  "id": "67ss0q0j5pth5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13658",
  "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",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221519225",
  "pdf_url": "https://arxiv.org/pdf/2610.13658",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/67ss0q0j5pth5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13658",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13658"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Regularization of a Divergent Arithmetic Product via the Asymptotic Monoid: The Case of $(-1)^{ω(n)}$",
  "title_text": "Regularization of a Divergent Arithmetic Product via the Asymptotic Monoid: The Case of (−1)^ω(n)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/67ss0q0j5pth5",
  "venue": "arXiv",
  "versions": []
}