Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian

We present asymptotically sharp inequalities for the eigenvalues mu(k) of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in [14]. For the Riesz mean R-1(z) of the eigenvalues we improve the known sharp semiclassical bound in terms of the volume of the domain with a second term with the best possible expected power of z. In addition, we obtain two-sided bounds for individual mu(k), which are semiclassically sharp, and we obtain a Neumann version of Laptev's result that the Polya conjecture is valid for domains that are Cartesian products of a generic domain with one for which Polya's conjecture holds. In a final section, we remark upon the Dirichlet case with the same methods.