{
  "abstract": "Yang introduced a tempered xi function $\\widehatξ(s)$ by replacing the hyperbolic cosine in a classical integral representation of the Riemann xi function by a hyperbolic sine, and conjectured that every zero of $\\widehatξ$ lies on the critical line $\\Re s=1/2$. We disprove this conjecture. After the change of variables $x=e^{2t}$, one has \\[ \\widehatξ\\!\\left(\\tfrac12+iz\\right)=iS(z), \\qquad S(z)=\\int_0^\\infty K(t)\\sin(zt)\\,dt, \\] where $K$ is positive, smooth, and doubly exponentially decaying. Two integrations by parts give $S(y)=K(0)/y+O(y^{-2})$ for real $|y|\\to\\infty$, with $K(0)>0$; hence $S$ has only finitely many real zeros. On the other hand $S$ is an entire function of order at most one. If it had only finitely many zeros in the complex plane, Hadamard factorization would force $S(z)=P(z)e^{az+b}$, which is incompatible with $S(y)\\to0$ as $y\\to\\pm\\infty$. Thus $S$ has infinitely many nonreal zeros, and consequently $\\widehatξ$ has infinitely many zeros off the critical line. The proof is unconditional and does not use the Riemann Hypothesis or numerical zero finding. A short numerical illustration is included.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29898",
  "author_total": 2,
  "authors": [
    {
      "name": "Aiken Kazin"
    },
    {
      "name": "Shirali Kadyrov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2dewe7zjxx86r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29898",
  "fulltext_url": null,
  "id": "2dewe7zjxx86r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29898",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214286963",
  "pdf_url": "https://arxiv.org/pdf/2609.29898",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2dewe7zjxx86r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29898",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29898"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Infinitely Many Off-Critical-Line Zeros of the Tempered Xi Function: A Disproof of Yang's Conjecture",
  "title_text": "Infinitely Many Off-Critical-Line Zeros of the Tempered Xi Function: A Disproof of Yang's Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/2dewe7zjxx86r",
  "venue": "arXiv",
  "versions": []
}