{
  "abstract": "Ono, Schneider and Wagner found a partition-theoretic analogue of Alladi's duality formula: for many sets $S\\subseteq\\N$, the series $-\\sumμ_{\\mathcal P}(λ)q^{|λ|}$ over partitions into distinct parts with smallest part in $S$ tends to the density of $S$ as $q\\to1$. We determine when this happens. Radially, the summation kernel is a Gumbel density, and Wiener's theorem shows that for bounded $f$ the associated series tends to $L$ if and only if $f$ has mean $L$ on every interval $(x,x+cx/\\log x]$; natural density does not suffice. Multiplied by $\\log(1/(1-q))$, the series tends to $δ$ if and only if $S$ has $(cδ+o(1))x/(\\log x)^2$ elements in each such interval; this gives a partition analogue of Alladi's theorem for Chebotarev sets of primes. Off the real axis the largest weights grow like $e^{Ψ_β/t}$ with $Ψ_β>0$. Along $q=e^{-t(1+iβ)}$, the formula for residue classes modulo $m\\ge3$ holds exactly when $|β|<β_m$, an explicit Clausen-function threshold with $β_m\\to0$, and it fails almost surely for random sets of bounded discrepancy. These results correct several published claims.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10570",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohammad F. Marashdeh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0w8186tcbe3ev.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10570",
  "fulltext_url": null,
  "id": "0w8186tcbe3ev",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10570",
  "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": "W7221105017",
  "pdf_url": "https://arxiv.org/pdf/2610.10570",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0w8186tcbe3ev.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10570",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10570"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Partition-theoretic density formulas: radial Tauberian theory and non-radial instability",
  "title_text": "Partition-theoretic density formulas: radial Tauberian theory and non-radial instability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/0w8186tcbe3ev",
  "venue": "arXiv",
  "versions": []
}