Riesz means of the distribution function of the eigenvalues of an elliptic operator

Let $N(\lambda)$ be the distribution function of eigenvalues of a elliptic operator. We investigate the asymptotics of the Eiesz means of $N(\lambda)$ when $\lambda\to\infty$. Under some assumptions (they are fulfilled, in particular, for Laplace operator) we obtaine the nest term of asymptotics in comparison with known formulas.