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.