Weyl Law Improvement for Products of Spheres

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.