Dirac functional determinants in terms of the eta invariant and the noncommutative residue

The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrinsic ambiguity present in the definition of the associated fermion functional determinant in the massless case and, also, the unavoidable presence (in some situations) of a multiplicative anomaly, that can be conveniently expressed in terms of the non-commutative residue. The ambiguity is here proven to disappear in the massive case, giving rise to a phase of the Dirac determinant - that agrees with very recent calculations which appeared in the mathematical literature - and to a multiplicative anomaly - also in agreement with other calculations, in the coinciding cases (in fact our results cover much more general situations). A number of physically relevant explicit examples are worked out. After explicit, nontrivial resummation of the mass series expansions, involving zeta and eta functions, our results are finally expressed in terms of quite simple formulae.