{
  "abstract": "Let $p_n$ be the $n$th prime and $g_n=p_{n+1}-p_n$. Following Gafni and Tao, an integer $m$ with $p_n<m<p_{n+1}$ is rough for the gap if its least prime factor is at least $g_n$; let $M_n$ be the number of such integers. Gafni and Tao proved that the number of $n$ with $X\\le p_n\\le 2X$ and $M_n=0$ is $O(X/\\log^2 X)$, and that it is asymptotic to $cX/\\log^2 X$ under a form of the Dickson-Hardy-Littlewood prime tuples conjecture. We study $N_r(X)$, the number of $n$ with $X\\le p_n\\le 2X$ and $M_n<r$. For every $r\\ge 1$ we define a constant $c_r$ by an endpoint-conditioned local sieve model, with $c_1=c$, and we prove unconditionally that $c_r=h_r^*+O_\\varepsilon(r^{1/2}(\\log r)^{1+\\varepsilon})$ for $r\\ge 3$, where $h_r^*$ is the least even $h\\ge 2$ with $h\\prod_{p<h}(1-1/p)\\ge r$. Consequently $c_r=H_r(1+O(e^{-κ\\sqrt{\\log r}}))$ for an absolute constant $κ>0$, where $H_r>e$ solves $e^{-γ}H_r/\\log H_r=r$ ($γ$ is Euler's constant), and $c_r=e^γr(\\log r+\\log\\log r+γ+(\\log\\log r+γ)/\\log r+O((\\log\\log r)^2/(\\log r)^2))$. The proof combines the Montgomery-Vaughan moment bounds for reduced residues in short intervals with an exact removal of the endpoint conditioning. For each fixed $r$, the argument of Gafni and Tao extends to give $N_r(X)\\ll_r X/\\log^2 X$ unconditionally, and $N_r(X)\\sim c_rX/\\log^2 X$ under their conjecture. Exact enumeration of the local model gives $c\\ge 2.7413$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10582",
  "author_total": 1,
  "authors": [
    {
      "name": "Bahar Uğurdoğan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ew4fz2mzrnsh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10582",
  "fulltext_url": null,
  "id": "7ew4fz2mzrnsh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10582",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221088922",
  "pdf_url": "https://arxiv.org/pdf/2610.10582",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ew4fz2mzrnsh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10582",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10582"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Multiplicity Constants for Rough Numbers in Prime Gaps",
  "title_text": "Multiplicity Constants for Rough Numbers in Prime Gaps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/7ew4fz2mzrnsh",
  "venue": "arXiv",
  "versions": []
}