  We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a
bounded domain with Lipschitz boundary and prove two-term asymptotics for their
Riesz means of arbitrary positive order. Moreover, when the underlying domain
is convex, we obtain universal, non-asymptotic bounds that correctly reproduce
the two leading terms in the asymptotics and depend on the domain only through
simple geometric characteristics. An important ingredient in the proof of the
latter result is a pointwise bound for the heat kernel of the Neumann Laplacian
in a convex domain with universal constants. Additional ingredients in our
proof are non-asymptotic versions of various Tauberian theorems.
