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 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. 
Kamil Mroz, Alexander Strohmaier
math.SP math.DG