{
  "abstract": "Let $G_n$ be the graph on $\\{2,\\ldots,n\\}$ in which two integers are adjacent when they have a common divisor greater than one. We determine the complete asymptotic expansion of its Havel--Hakimi residue $\\R(G_n)$, confirming a leading-constant prediction of Staton recorded in Fajtlowicz's \\emph{Written on the Wall}. If $A=\\sum_{k=2}^{\\infty}(\\log k)/(k^2(k-1))$, then the first two terms are $\\R(G_n)=(ζ(2)-1)n/\\log n+(ζ(2)-1-A)n/\\log^2n +O(n/\\log^3n)$. More precisely, the difference between $\\R(G_n)$ and the prime-vertex contribution to the Caro--Wei sum is $O_β(n\\exp\\{-(\\log n)^β\\})$ for every fixed $0<β<1/2$; this estimate yields every coefficient in the expansion. The upper bound follows from a degree-preserving realization in which almost all relevant prime vertices are partitioned into cliques. We also prove that the unlabeled graph determines $π(n)$ through its simplicial true-twin classes. Stable inverses for weighted sums of the resulting degree-class counts give criteria equivalent to the Riemann hypothesis, including one involving only the Caro--Wei sum. An exact local-defect identity additionally reduces the conjectured sharp $+2$ residue bound to explicit prefix estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.04040",
  "author_total": 1,
  "authors": [
    {
      "name": "Randy Davila"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/be99fatg5fhq6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.04040",
  "fulltext_url": null,
  "id": "be99fatg5fhq6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.04040",
  "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.CO"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7196947220",
  "pdf_url": "https://arxiv.org/pdf/2608.04040",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/be99fatg5fhq6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.04040",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.04040"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure",
  "title_text": "Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/be99fatg5fhq6",
  "venue": "arXiv",
  "versions": []
}