Generalized Krein formula, determinants and Selberg zeta function in even dimension

39 pages. Exposition improved. Weyl asymptotic precisedInternational audienceFor a class of even dimensional conformally compact manifolds (X,g), we define a generalized Krein spectral function by applying a renormalized trace functional to the spectral measure of the Laplacian. We then show that this is the phase of the Kontsevich-Vishik determinant det S(s) of the scattering operator S(s) of (the Laplacian of) g and we analyze the divisors of det(S(s)). As an application for convex co-compact hyperbolic quotients, we obtain a functional equation for Selberg's Zeta function Z(s), we express the determinant of the GJMS conformal Laplacians of the conformal infinity of (X,g) in term of particular values of Z(s), and we show a sharp Weyl type asymptotic for the Krein function