{
  "abstract": "In this article, we prove the irrationality of at least one of the three numbers $ζ(5), ζ(7)$ and $ζ(9)$ is irrational. We find linear forms in $1,ζ(5),ζ(7)$ and $ζ(9)$ and prove their arithmetic by Kranttenthaler and Rivoal. We also refine these estimates by using integral quantity $Φ$ involving specific product of prime powers. We use the results of Kranttenthaler, Rivoal and Zudilin to prove that at least one of the three numbers $ζ(5), ζ(7)$ and $ζ(9)$ is irrational.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22316",
  "author_total": 1,
  "authors": [
    {
      "name": "Shekhar Suman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19v83678t2hrn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22316",
  "fulltext_url": null,
  "id": "19v83678t2hrn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22316",
  "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|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214045719",
  "pdf_url": "https://arxiv.org/pdf/2609.22316",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19v83678t2hrn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22316",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22316"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Advanced Mathematical Identities",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "At least one of the three numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ is irrational",
  "title_text": "At least one of the three numbers ζ(5), ζ(7), ζ(9) is irrational",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/19v83678t2hrn",
  "venue": "arXiv",
  "versions": []
}