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Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts
Abstract
We study the number NI(L) of zeros of the Hardy--Szegő zero process in the horizontal window [0,L]× I as L→∞, where I=[α,β]Subset(0,∞) 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≥0, we determine a full asymptotic formula for ℙ{NI(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 bL≤ n≤εL L, where bL→∞, εL↓0, and bL≤εL L, identifying the large-deviation exponent, endpoint correction, and Gaussian prefactor.
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BibTeX
@misc{chen2026endpoint,
title = {{Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts}},
author = {Bonan Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08991},
eprint = {2609.08991},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08991},
}
RIS
TY - UNPB
TI - Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts
AU - Bonan Chen
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08991
UR - https://arxiv.org/abs/2609.08991
AB - We study the number N_I(L) of zeros of the Hardy--Szegő zero process in the horizontal window [0,L]× I as L→∞, where I=[α,β]Subset(0,∞) 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≥0, we determine a full asymptotic formula for 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≤ n≤ε_L L, where b_L→∞, ε_L↓0, and b_L≤ε_L L, identifying the large-deviation exponent, endpoint correction, and Gaussian prefactor.
ER -