A note on the resonance counting function for surfaces with cusps

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 $T\to \mathrm{\infty }$ with remainder $O(T^{3\mathrm{/}2})$.