We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in ${\mathbb{R}}^2$. Unlike other lattice point problems, the one arisen naturally here has interesting features that lattice points under consideration are translated by various amounts and the curvature of the boundary is unbounded. By transforming this problem into a relatively standard form and using classical van der Corput's bounds, we obtain a two-term Weyl formula for the eigenvalue counting function for the planar annulus with a remainder of size $O({\mu }^{2/3})$. If we additionally assume that certain tangent has rational slope, we obtain an improved remainder estimate of the same strength as Huxley's bound in the Gauss circle problem, namely $O({\mu }^{131/208}{(\log \mu )}^{18627/8320})$. As a by-product of our lattice point counting results, we readily obtain this Huxley-type remainder estimate in the two-term Weyl formula for planar disks.

我们研究Bessel函数的叉积的零点并获得它们的近似值，在此基础上，我们将与平面环空相关联的Dirichlet Laplacian的特征值计数问题简化为与一个特定域相关的格点计数问题。 ${\mathbb{[R}}^{\mathrm{2个}}$。与其他晶格点问题不同，此处自然产生的问题具有有趣的特征，即所考虑的晶格点可以以不同的量平移，并且边界的曲率不受限制。通过将此问题转换为相对标准的形式并使用经典范德·科珀特的边界，我们获得了具有余下大小的平面环的特征值计数函数的两项Weyl公式$Ø（{\mu }^{\mathrm{2个}/3}）$。如果我们另外假设某些切线具有合理的斜率，我们将获得与高斯圆问题中的赫x黎界具有相同强度的改进余数估计，即$Ø（{\mu }^{131/208}{（\mathrm{日志}\mu ）}^{18627/8320}）$。作为格点计数结果的副产品，我们很容易在平面磁盘的两项Weyl公式中获得此Huxley型余数估计。