Pólya's eigenvalue conjecture is false for spheres

By comparing the Laplace spectrum of the sphere ${\mathbb{S}}^n$ to its Weyl function $w(x)=\frac{{\omega }_n}{(2\pi {)}^n}\mathrm{\mid }{\mathbb{S}}^n\mathrm{\mid }{x}^{n\mathrm{/}2}$, we show that no analogue of Pólya’s eigenvalue conjecture holds in general for Riemannian manifolds with positive sectional curvature.