  Fermion N-loops with an arbitrary number of density vertices N > d+1 in d
spatial dimensions can be expressed as a linear combination of (d+1)-loops with
coefficients that are rational functions of external momentum and energy
variables. A theorem on symmetrized products then implies that divergencies of
single loops for low energy and small momenta cancel each other when loops with
permuted external variables are summed. We apply these results to the
one-dimensional Fermi gas, where an explicit formula for arbitrary N-loops can
be derived. The symmetrized N-loop, which describes the dynamical N-point
density correlations of the 1D Fermi gas, does not diverge for low energies and
small momenta. We derive the precise scaling behavior of the symmetrized N-loop
in various important infrared limits.
