Density and location of resonances for convex co-compact hyperbolic surfaces

Let X = Gamma\H-2 be a convex co-compact hyperbolic surface and let delta be the Hausdorff dimension of the limit set. Let Delta(X) be the hyperbolic Laplacian. We show that the density of resonances of the Laplacian Delta(X) in rectangles {sigma <= Re(s) <= delta, vertical bar Im(s)vertical bar <= T} is less than O(T1+tau(sigma)) in the limit T -> infinity, where tau(sigma) < delta as long as sigma > delta/2. This improves the previous fractal Weyl upper bound of Zworski (Invent. Math. 136(2):353-409, 1999) and goes in the direction of a conjecture stated in Jakobson and Naud (Geom. Funct. Anal. 22(2):352-368, 2012).