  We present a new multiparameter resolvent trace expansion for elliptic
operators, polyhomogeneous in both the resolvent and auxiliary variables. For
elliptic operators on closed manifolds the expansion is a simple consequence of
the parameter dependent pseudodifferential calculus. As an additional
nontrivial toy example we treat here Sturm-Liouville operators with separated
boundary conditions.
  As an application we give a new formula, in terms of regularized sums, for
the zeta-determinant of an infinite direct sum of Sturm-Liouville operators.
The Laplace-Beltrami operator on a surface of revolution decomposes into an
infinite direct sum of Sturm-Louville operators, parametrized by the
eigenvalues of the Laplacian on the cross-section. We apply the polyhomogeneous
expansion to equate the zeta-determinant of the Laplace-Beltrami operator as a
regularized sum of zeta-determinants of the Sturm-Liouville operators plus a
locally computable term from the polyhomogeneous resolvent trace asymptotics.
This approach provides a completely new method for summing up zeta-functions of
operators and computing the meromorphic extension of that infinite sum to s=0.
  We expect out method to extend to a much larger class of operators.
