Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature

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 Bérard can be used to prove a logarithmic improvement for the remainder term of the Riesz means of the counting function of the Laplace operator.