On the zeta functions of prehomogeneous vector spaces for a pair of simple algebras

In this paper we consider the prehomogeneous vector space for a pair of simple algebras which are inner forms of the D-4 type and the E-6 type. We mainly study the non-split cases. The main purpose of this paper is to determine the principal parts of the global zeta functions associated with these spaces when the simple algebras are non-split. We also give a description of the sets of rational orbits of these spaces, which clarifies the expected density theorems arising from the properties of these zeta functions.