{
  "abstract": "This paper establishes an equivalence between the Riemann hypothesis and the association, together with uniform $L^2$-boundedness, of a moderate net in a Colombeau-type algebra built from polynomially damped Báez-Duarte sums. The regularization is performed by multiplicative (Mellin) convolution, which respects the dilation symmetry of the Beurling functions and guarantees that every approximant lies in the $L^2$-closure of the Beurling space. The equivalence is unconditional under the Riemann hypothesis: it uses only the qualitative convergence of Báez-Duarte, Mazur's theorem, and the classical Nyman--Beurling criterion. As a separate quantitative refinement, we prove that under two additional hypotheses on the non-trivial zeros of $ζ$ (simplicity and separation), the damping error admits a power-law bound with an explicit constant. This refinement is independent of the equivalence and is not used in its proof. The exponential damping $e^{-k\\eps^2}$ is discussed as an open problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.22562",
  "author_total": 2,
  "authors": [
    {
      "name": "Amaury Alvarez Cruz"
    },
    {
      "name": "Esteban A Alvarez Gutierrez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a1sy0tg8s3hzp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.22562",
  "fulltext_url": null,
  "id": "a1sy0tg8s3hzp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.22562",
  "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.CV"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7165673195",
  "pdf_url": "https://arxiv.org/pdf/2606.22562",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a1sy0tg8s3hzp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2606.22562",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.22562"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Multiplicative Colombeau algebras and the Nyman--Beurling criterion for the Riemann hypothesis",
  "title_text": "Multiplicative Colombeau algebras and the Nyman--Beurling criterion for the Riemann hypothesis",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/a1sy0tg8s3hzp",
  "venue": "arXiv",
  "versions": []
}