For compact Riemannian manifolds all of whose geodesics are closed (aka Zoll manifolds) one can define the determinant of a
zeroth order pseudodifferential operator by mimicking Szego’s definition of this determinant for the operator: multiplication by a
bounded function, on the Hilbert space of square-integrable functions on the circle. In this paper we prove that the non-local
contribution to this determinant can be computed in terms of a much simpler “zeta-regularized” determinant.