  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
amount 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.
