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Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts — arXiv:2609.08991

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.08991
License (as archived): CC0-1.0
Archived: 2026-10-16T14:35:22.041Z
SHA-256 of the archived text: 65bb9332d71712a5e8c4d627fc9e0ef951fab18e00d77f045186a91ae1bc6ca6
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How to cite

Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts — arXiv:2609.08991 — arXiv (en), https://arxiv.org/abs/2609.08991 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_6ca80eec8a79 (Metraipolis archive dc_6ca80eec8a79, sha256 65bb9332d71712a5e8c4d627fc9e0ef951fab18e00d77f045186a91ae1bc6ca6)

BibTeX

@misc{dc6ca80eec8a79,
  title = {Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts — arXiv:2609.08991},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.08991},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_6ca80eec8a79; sha256 65bb9332d71712a5e8c4d627fc9e0ef951fab18e00d77f045186a91ae1bc6ca6},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.08991. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 \(I=[\alpha,\beta]\Subset(0,\infty)\) 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.PR).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2609.08991 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:09:59.815Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).