{
  "abstract": "In this paper, we consider the asymptotic density of $\\#\\{p\\leq x:P^+(p-1)\\geq p^c\\}$ and $\\#\\{n\\leq x:P^+(n)<P^+(n+1)\\}$, where $P^+(n)$ denote the largest prime factor of $n$. We show that for $x\\rightarrow\\infty$, one has \\begin{align*} \\mathop{\\lim\\sup}_{x\\rightarrow\\infty} \\frac{1}{π(x)}\\#\\{p\\leq x:P^+(p-1)\\geq p^c\\}\\leq\\frac{16}{5}\\log\\frac{1}{c} \\end{align*} where $e^{-5/16}<c<1$, and \\begin{align*} \\#\\{n\\leq x:P^+(n)<P^+(n+1)\\}>0.299x. \\end{align*} The first result constitutes an improvement upon that of Ding and Wang (2025), who obatined $\\mathop{\\lim\\sup}_{x\\rightarrow\\infty} \\frac{1}{π(x)}\\#\\{p\\leq x:P^+(p-1)\\geq p^c\\}\\leq \\frac{7}{2}\\log\\frac{1}{c}$. The second result improves a previous result $0.280$ by the author (2026). The key to the proof is that for a special class of convolution forms equipped with well-factorable weights, we may use the level \\(x^{5/8-o(1)}\\) for Pascadi's prime-distribution result with triple-well-factorable weights. We also use Pascadi's estimation of incomplete Kloosterman sums.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.13299",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhiyuan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/da0w6j3emzc38.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.13299",
  "fulltext_url": null,
  "id": "da0w6j3emzc38",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.13299",
  "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": "W7203459309",
  "pdf_url": "https://arxiv.org/pdf/2608.13299",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/da0w6j3emzc38.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.13299",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.13299"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications",
  "title_text": "Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/da0w6j3emzc38",
  "venue": "arXiv",
  "versions": []
}