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

RESUMO

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.