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

Source 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.