  For a general Fuchsian group of the first kind with an arbitrary unitary
representation we define zeta functions related to the contributions of the
identity, hyperbolic, elliptic and parabolic conjugacy classes in Selberg's
trace formula. We present Selberg's zeta function in terms of a regularized
determinant of the automorphic Laplacian. We also present the zeta function for
the identity contribution in terms of a regularized determinant of the
Laplacian on the two dimensional sphere. We express the zeta functions for the
elliptic and parabolic contributions in terms of certain regularized
determinants of one dimensional Schroedinger operator for harmonic oscillator.
We decompose the determinant of the automorphic Laplacian into a product of the
determinants where each factor is a determinant representation of a zeta
function related to Selberg's trace formula. Then we derive an identity
connecting the determinants of the automorphic Laplacians on different
Riemannian surfaces related to the arithmetical groups. Finally, by using the
Jacquet-Langlands correspondence we connect the determinant of the automorphic
Laplacian for the unit group of quaternions to the product of the determinants
of the automorphic Laplacians for certain cogruence subgroups.
