### Abstract

We obtain asymptotic lower bounds for the spectral function of the Laplacian on compact manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Our results can be considered pointwise versions (on a general manifold) of Hardy's lower bounds for the error term in the Gauss circle problem.

Next, we obtain a lower bound for the remainder in Weyl's law on a negatively curved surface. On higher-dimensional negatively curved manifolds, we prove a similar bound for the oscillatory error term. This extends earlier results of Hejhal and Randol on surfaces of constant negative curvature proved by methods of analytic number theory. Our approach uses wave trace asymptotics, equidistribution of closed geodesics and small-scale microlocalization.

### Related Links

- http://www.ams.org/era/2005-11-09/S1079-6762-05-00149-6/home.html - ERA-AMS paper
- http://front.math.ucdavis.edu/math.SP/0505400 - arxiv preprint
- http://www.math.mcgill.ca/jakobson/papers/umd06.pdf - Slides