{
  "abstract": "Let $P$ be a trigonometric polynomial with only real zeros. We prove that at most $\\frac{2}{3}$ of its zeros coincide with the zeros of the Riemann zeta function $ζ(s)$ on the critical line. We also show that \\begin{equation*} \\sup_{\\substack{T\\leq x\\leq2T\\\\ P(x)=0}} \\log \\left|ζ\\left(\\frac{1}{2}+ix\\right)\\right| \\geq (1+o(1))\\sqrt{\\frac{\\log T}{6\\log\\log T}}. \\end{equation*} These results extend the corresponding theorems of Li and Radziwill on the zeros of $ζ(s)$ along vertical arithmetic progressions. The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure $\\sum_{P(γ)=0} {\\bf δ}_γ$ is crystalline.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26705",
  "author_total": 4,
  "authors": [
    {
      "name": "Felipe Gonçalves"
    },
    {
      "name": "Bryce Kerr"
    },
    {
      "name": "Oleksiy Klurman"
    },
    {
      "name": "Guilherme Vedana"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a0vpkr50bz32p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26705",
  "fulltext_url": null,
  "id": "a0vpkr50bz32p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26705",
  "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.CA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223044472",
  "pdf_url": "https://arxiv.org/pdf/2610.26705",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a0vpkr50bz32p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.26705",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26705"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Riemann zeta function at the zeros of real-rooted trigonometric polynomials",
  "title_text": "The Riemann zeta function at the zeros of real-rooted trigonometric polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/a0vpkr50bz32p",
  "venue": "arXiv",
  "versions": []
}