  We analyze the original proof by Luttinger and Ward of the Luttinger theorem,
according to which for uniform ground states of systems of (interacting)
fermions, which may be metallic or insulating, the number of k points
corresponding to non-negative values of G_s(k;mu) is equal to the total number
of particles with spin index s in these ground states. Here G_s(k;mu) is the
single-particle Green function of particles with spin index s at the chemical
potential mu. For the cases where the two-body interaction potential is
short-range, and in particular for lattice models, we explicitly demonstrate
that this theorem is unconditionally valid, irrespective of the strength of the
bare interaction potential. We arrive at this conclusion by amongst other
things demonstrating that the perturbation series expansion for self-energy in
terms of skeleton diagrams, as encountered in the proof of the Luttinger-Ward
identity, is uniformly convergent for almost all momenta and energies. We
further investigate the mechanisms underlying some reported instances of
failure of the Luttinger theorem. With one exception, for all the cases
considered in this paper, we show that the apparent failures of the Luttinger
theorem can be attributed either to shortcomings of the employed
single-particle Green functions or to misapplication of this theorem. The one
exceptional case brings to light the possibility of a genuine failure of the
Luttinger theorem for insulating ground states, which we show to be brought
about by a false limit that in principle can be reached on taking the
zero-temperature limit without the value of mu coinciding with the
zero-temperature limit of the chemical potential satisfying the equation of
state at finite temperatures; no such ambiguity can arise for metallic states.
