{
  "abstract": "The Riemann zeta function ζ(s) is central to analytic number theory and the study of prime distribution. Its non-trivial zeros, located in the critical strip 0 < ℜ(s) < 1, are of fundamental importance. While the Riemann Hypothesis concerns their location on the critical line ℜ(s) = 1 2 , another deep question concerns their simplicity: whether each zero is of multiplicity one. Recent works (Alpöge-Furman 2026, Lamzouri 2026, Goldston et al. 2025, Chavez 2024) have established that a large proportion of zeros are simple, or provided conditional results under conjectures. In this paper, has been presented a new unconditional proof that every non-trivial zero of ζ(s) is simple. The presented approach uses integral representations I(s) and I ′ (s), a reductio ad absurdum argument showing they cannot coincide in the critical strip, and de l'Hôpital's principle at a zero. This analytic framework leads to a contradiction if a multiple zero exists, thereby establishing the simplicity of all non-trivial zeros.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Mansour Salem, S Hashad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4y8qkcw8sq5e5.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "4y8qkcw8sq5e5",
  "issns": [],
  "landing_url": "https://hal.science/hal-05751250v1",
  "language": "en",
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05751250/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4y8qkcw8sq5e5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:35Z",
      "native_id": "hal-05751250",
      "source": "hal",
      "url": "https://hal.science/hal-05751250v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "A Direct Proof of the Simplicity of Non-Trivial Zeros of the Riemann Zeta Function",
  "title_text": "A Direct Proof of the Simplicity of Non-Trivial Zeros of the Riemann Zeta Function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:35Z",
  "url": "https://research.draconitic.com/paper/4y8qkcw8sq5e5",
  "venue": "HAL",
  "versions": []
}