{
  "abstract": "We study the number \\(N_I(L)\\) of zeros of the Hardy--Szegő zero process in the horizontal window \\([0,L]\\times I\\) as \\(L\\to\\infty\\), where \\(I=[α,β]\\Subset(0,\\infty)\\) is fixed. We obtain sharp point-probability asymptotics at the lower endpoint of the density scale. In the fixed-count regime, for every fixed integer \\(k\\geq0\\), we determine a full asymptotic formula for \\(\\mathbb P\\{N_I(L)=k\\}\\), identifying its exponential rate, order-one correction, and \\(k\\)-dependent polynomial prefactor; the case \\(k=0\\) gives the hole probability. In the vanishing-density regime, we prove a uniform local asymptotic formula for \\(b_L\\leq n\\leq\\varepsilon_L L\\), where \\(b_L\\to\\infty\\), \\(\\varepsilon_L\\downarrow0\\), and \\(b_L\\leq\\varepsilon_L L\\), identifying the large-deviation exponent, endpoint correction, and Gaussian prefactor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08991",
  "author_total": 1,
  "authors": [
    {
      "name": "Bonan Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4a985hnbrnamd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08991",
  "fulltext_url": null,
  "id": "4a985hnbrnamd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08991",
  "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.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7212041997",
  "pdf_url": "https://arxiv.org/pdf/2609.08991",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4a985hnbrnamd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08991",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08991"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts",
  "title_text": "Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/4a985hnbrnamd",
  "venue": "arXiv",
  "versions": []
}