  We investigate the validity of Luttinger's theorem (or Luttinger sum rule) in
two scale-invariant fermionic models. We find that, in general, Luttinger's
theorem does not hold in a system of fermions with power-law Green functions
which do not necessarily preserve particle-hole symmetry. However, Ref.
\cite{Blagoev1997,Yamanaka1997} showed that Luttinger liquids, another
scale-invariant fermionic model, respect Luttinger's theorem. To understand the
difference, we examine the spinless Luttinger liquid model. We find two
properties which make the Luttinger sum rule valid in this model: particle-hole
symmetry and $\mathrm{Im} G(\omega=0,-\infty)=0$. We conjecture that these two
properties represent sufficient, but not necessary, conditions for the validity
of the Luttinger sum rule in condensed matter systems.
