  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 \cite{HaSt14}. 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. In a final section, we remark upon the Dirichlet case
with the same methods.
