The Weyl formula for planar annuli

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