Endpoint and Vanishing-Density Asymptotics for Hardy--Szegő Zero Counts

We study the number $N_I(L)$ of zeros of the Hardy–Szegő zero process in the horizontal window $[0,L]\times I$ as $L\to \mathrm{\infty }$, where $I=[\alpha ,\beta ]⋐(0,\mathrm{\infty })$ 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 $k\ge 0$, 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 $b_L\le n\le {\epsilon }_{L}L$, where $b_L\to \mathrm{\infty }$, ${\epsilon }_L\downarrow 0$, and $b_L\le {\epsilon }_{L}L$, identifying the large-deviation exponent, endpoint correction, and Gaussian prefactor.