  We prove some sufficient conditions in order that a root of the
Bernstein-Sato polynomial contributes to a difference between certain D-modules
generated by rational powers of a holomorphic function; for instance, this
holds in the case of isolated singularities with semisimple Milnor monodromies.
We then construct an example where a root does not contribute to a difference.
This also solves an old open problem about the relation between the Milnor
monodromy and the exponential of the residue of the Gauss-Manin connection on
the saturation of the Brieskorn lattice. This shows that the structure of
Brieskorn lattices can be more complicated than one might imagine.
