  Let $(M, {g})$ be a compact, $d$-dimensional Riemannian manifold without
boundary. Suppose further that $(M,g)$ is either two dimensional and has no
conjugate points or $(M,g)$ has non-positive sectional curvature. The goal of
this note is to show that the long time parametrix obtained for such manifolds
by B\'erard can be used to prove a logarithmic improvement for the remainder
term of the Riesz means of the counting function of the Laplace operator.
