{
  "abstract": "We study a divisor-weighted version of Golomb's self-description: the nondecreasing sequence $A$ in which each positive integer $m$ occurs $\\sum_{d\\mid m}A(d)$ times. This rule has a unique solution, and $$ A(n)\\sim Cn^{φ-1},\\quad C=\\left(\\fracφ{ζ(φ)}\\right)^{2-φ},\\quad φ=\\frac{1+\\sqrt{5}}{2}. $$ The golden-ratio exponent of Golomb's sequence survives, while divisor weighting changes the leading constant. We prove the asymptotic without assuming regular variation, by a contraction of upper and lower power envelopes. The argument applies to every nonnegative integer Dirichlet-convolution kernel $w$ with $w(1)=1$ and $\\sum_q w(q)q^{-φ}<\\infty$. For the divisor kernel, a quantitative version gives relative error $O_γ((\\log n)^{-γ})$ for every $0<γ<1$. Although the global growth is smooth, the normalized run lengths retain arithmetic fluctuations. A uniform comparison with the divisor sum $\\sum_{d\\mid m}d^{1-φ}$ transfers them to an explicit random Euler product. We prove that its law is singular continuous with full support $[1,\\infty)$, determine all real Mellin moments, obtain the sharp maximal order of the fluctuations, and show that sampling sequence positions instead of block labels produces the size-biased law.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03811",
  "author_total": 1,
  "authors": [
    {
      "name": "Marco Mantovanelli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5skhjze09nwkv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03811",
  "fulltext_url": null,
  "id": "5skhjze09nwkv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03811",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220665031",
  "pdf_url": "https://arxiv.org/pdf/2610.03811",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5skhjze09nwkv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03811",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03811"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Divisor-Weighted Golomb Sequence: Zeta Asymptotics and Arithmetic Fluctuations",
  "title_text": "A Divisor-Weighted Golomb Sequence: Zeta Asymptotics and Arithmetic Fluctuations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/5skhjze09nwkv",
  "venue": "arXiv",
  "versions": []
}