  Two different methods for establishing a space-like Coulomb sum rule for the
relativistic Fermi gas are compared. Both of them divide the charge response by
a normalizing factor such that the reduced response thus obtained fulfills the
sum rule at large momentum transfer. To determine the factor, in the first
approach one exploits the scaling property of the longitudinal response
function, while in the second one enforces the completeness of the states in
the space-like domain via the Foldy-Wouthuysen transformation. The
energy-weighted and the squared-energy-weighted sum rules for the reduced
responses are explored as well and the extension to momentum distributions that
are more general than a step-function is also considered. The two methods yield
reduced responses and Coulomb sum rules that saturate in the non-Pauli-blocked
region, which can hardly be distinguished for Fermi momenta appropriate to
atomic nuclei. Notably the sum rule obtained in the Foldy-Wouthuysen approach
coincides with the well known non-relativistic one. Only at quite large
momentum transfers (say 1 GeV/c) does a modest softening of the
Foldy-Wouthuysen reduced response with respect to that obtained in the scaling
framework show up. The two responses have the same half-width to second order
in the Fermi momentum expansion. However, when distributions extending to
momenta larger than that at the Fermi surface are employed, then in both
methods the Coulomb sum rule saturates only if the normalizing factors are
appropriately modified to account for the high momentum components of the
nucleons.
