{
  "abstract": "The real-axis jump of the prime-zeta function gives an integral form of Gram's series for Riemann's $R$-function. Applying the same operation to the classical almost-prime generating functions defines a smooth family $R_k$, with $R_1=R$. We give the construction and a convergence argument, and explain why its expansion at every fixed logarithmic order is the Selberg--Delange expansion. The distinction lies in the smaller real-axis contributions retained by the definition, rather than in new asymptotic coefficients. A finite Perron decomposition also identifies the additional contour terms required for exact recovery of the count.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11024",
  "author_total": 1,
  "authors": [
    {
      "name": "Martin Belton"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4aes1jydn2qgw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11024",
  "fulltext_url": null,
  "id": "4aes1jydn2qgw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11024",
  "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": "W7221072282",
  "pdf_url": "https://arxiv.org/pdf/2610.11024",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4aes1jydn2qgw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11024",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11024"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A note on Riemann's $R$-function and almost primes",
  "title_text": "A note on Riemann's R-function and almost primes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/4aes1jydn2qgw",
  "venue": "arXiv",
  "versions": []
}