A note on the resonance counting function for surfaces with cusps

Abstract

We prove sharp upper bounds for the number of resonances in boxes of size 1 at high frequency for the Laplacian on finite volume surfaces with hyperbolic cusps. As a corollary, we obtain a Weyl asymptotic for the number of resonances in balls of size $T\to \infty$ with remainder $O(T^{3/2})$.