This is a joint work with Victor Guillemin. We compare two ways of regularizing the determinant of a pseudo-differential operator of zeroth order. The first way is based on a generalization of Szego's strong limit theorem; the second regularization uses a generalized zeta-function. Both regularizations of the determinant are non-local, and they give different answers. However, and this is our main result, the ratio of them is a local quantity.