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Abstract. We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on H n+1 : in strips parallel to the imaginary axis the zeta function is bounded by exp(C|s| δ ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp(C|s| n+1 ), and it gives new bounds on the number of resonances (scattering poles) of Γ\H n+1 . The proof of this result is based on the application of holomorphic L 2 -techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider Γ\H n+1 as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic L 2 -techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets.

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