  For 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.
