{
  "abstract": "By a theorem of Bagchi, the Riemann Hypothesis (RH) holds if and only if $E(s) = 1/s$ belongs to the closed linear span of $G_k(s) = (k^{-s} - k^{-1})ζ(s)/s$, $k \\geq 2$, in the Hardy space $H^2(\\mathbb{C}_{1/2})$, where $\\mathbb{C}_α$ is the half-plane $\\mathrm{Re}(s) > α$. The formal Möbius expansion $E = \\sum_{k \\geq 2} μ(k) G_k$ diverges on the critical line, by a theorem of Báez-Duarte. We study it in the scale of spaces $H^2(\\mathbb{C}_α)$, where its convergence forces $ζ$ to be zero-free in $\\mathbb{C}_α$. We prove that the partial sums $S_n$ converge to $E$ in $H^2(\\mathbb{C}_α)$ for every $α> 1$, and that a single zero of $ζ$ in $\\mathbb{C}_α$ keeps the best-approximation error $d_n(α)^2$ uniformly away from zero. We express the Möbius error $s_n(α) = \\|S_n - E\\|^2_α$ as a series with elementary terms, free of $ζ$, and compute it for $n \\leq 10^6$: $s_n(1/2)$ oscillates without decaying, whereas for $α> 1/2$ it decays at a rate close to $n^{1-2α}$, much faster than the bound known under RH. On the critical line, $d_n(1/2)^2$ behaves like $0.046/\\log n$ for $100 \\leq n \\leq 5120$, in agreement with earlier computations by Báez-Duarte, Balazard, Landreau and Saias.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03924",
  "author_total": 2,
  "authors": [
    {
      "name": "Juan Manzur"
    },
    {
      "name": "Gustavo Quintero"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19ah4myd6g44z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03924",
  "fulltext_url": null,
  "id": "19ah4myd6g44z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03924",
  "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.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03924",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19ah4myd6g44z.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03924",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03924"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Hardy space approximation supporting zero-free half-planes for the $ζ$-function",
  "title_text": "A Hardy space approximation supporting zero-free half-planes for the ζ-function",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/19ah4myd6g44z",
  "venue": "arXiv",
  "versions": []
}