A lower bound for the remainder in Weyl's law on negatively curved surfaces

Abstract

We obtain an estimate from below for the remainder in Weyl's law on negatively curved surfaces. In the constant curvature case, such a bound was proved independently by Hejhal and Randol in 1976 using the Selberg zeta function techniques. Our approach works in arbitrary negative curvature, and is based on wave trace asymptotics for long times, equidistribution of closed geodesics and small-scale microlocalization.