  We give an improvement of sharp Berezin type bounds on the Riesz means
$\sum_k(\Lambda-\lambda_k)_+^\sigma$ of the eigenvalues $\lambda_k$ of the
Dirichlet Laplacian in a domain if $\sigma\geq 3/2$. It contains a correction
term of the order of the standard second term in the Weyl asymptotics. The
result is based on an application of sharp Lieb-Thirring inequalities with
operator valued potential to spectral estimates of the Dirichlet Laplacian in
domains.
