  We give a new upper bound on the Selberg zeta function for a convex
co-compact Schottky group acting on $ {\mathbb H}^{n+1}$: in strips parallel to
the imaginary axis the zeta function is bounded by $ \exp (C |s|^\delta) $
where $ \delta $ 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 $ \Gamma
\backslash {\mathbb 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 $ \Gamma \backslash {\mathbb 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.
