  The classical Weyl Law says that if $N_M(\lambda)$ denotes the number of
eigenvalues of the Laplace operator on a $d$-dimensional compact manifold $M$
without a boundary that are less than or equal to $\lambda$, then $$
N_M(\lambda)=c\lambda^d+O(\lambda^{d-1}).$$
  In this paper, we show Duistermaat and Guillemin's result allows us to
replace the $O(\lambda^{d-1})$ error with $o(\lambda^{d-1})$ if $M$ is a
product manifold. We quantify this bound in the case of Cartesian product of
spheres by reducing the problem to the study of the distribution of weighted
integer lattice points in Euclidean space and formulate a conjecture in the
general case reminiscent of the sum-product phenomenon in additive
combinatorics.
