Let be a compact, -dimensional Riemannian manifold without boundary. Suppose further that is either two dimensional and has no conjugate points or 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.
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Kamil Mroz, Alexander Strohmaier, Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature. J. Spectr. Theory 6 (2016), no. 3, pp. 629–642DOI 10.4171/JST/134