{
  "abstract": "We prove strict log-concavity of the positive functions $Θ$ and $Λ$ with one-sided and bilateral Laplace transforms $1/ξ(\\tfrac12+\\sqrt s)$ and $1/Ξ$, respectively, where $Ξ(z)=ξ(\\tfrac12+iz)$. For either function, log-concavity is the Pólya frequency property of order two (PF$_2$). Without assuming the Riemann hypothesis (RH), but using its published verification up to height about $3\\cdot10^{12}$, we further prove that both densities are PF$_{k_*}$ for $k_*=1\\,810\\,540\\,931\\,075$. The proof views $Θ$ as a small signed perturbation of a density that is PF of every order, with an error bound independent of the order, so the numerical check reduces to scalar inequalities. Since $Λ$ is a Gaussian mixture of $Θ$, it inherits the same order. Conversely, a pole obstruction shows that if either density is PF$_k$, then every zeta zero lower than the $(k+2)$-nd lies on the critical line. Finally, we prove that if RH holds up to height $T>\\tfrac12$, then $Ξ$ satisfies the generalized Laguerre inequalities $L_n[Ξ](x)\\ge0$ of every order $n$ for $|x|\\le T-\\tfrac12$, extending the case $n=1$ of Csordas, Ruttan and Varga (1991).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02944",
  "author_total": 1,
  "authors": [
    {
      "name": "Suvrit Sra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eyth37sq8w0ed.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02944",
  "fulltext_url": null,
  "id": "eyth37sq8w0ed",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02944",
  "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.CA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02944",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eyth37sq8w0ed.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.02944",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02944"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On some positivity properties of the Riemann $Ξ$ function",
  "title_text": "On some positivity properties of the Riemann Ξ function",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/eyth37sq8w0ed",
  "venue": "arXiv",
  "versions": []
}