  We consider a quantum multi-component plasma made with S species of point
charged particles interacting via the Coulomb potential. We derive the screened
activity series for the pressure in the grand-canonical ensemble within the
Feynman-Kac path integral representation of the system in terms of a classical
gas of loops. This series is useful for computing equations of state for it is
non-perturbative with respect to the strength of the interaction and it
involves relatively few diagrams at a given order. The known screened activity
series for the particle densities can be recovered by differentiation. The
particle densities satisfy local charge neutrality thanks to a Debye-screening
dressing mechanism of the diagrams in these series. We introduce a new general
neutralization prescription, based on this mechanism, for deriving approximate
equations of state where consistency with electroneutrality is automatically
ensured. This prescription is compared to other ones, including a
neutralization scheme inspired by the Lieb-Lebowitz theorem and based on the
introduction of (S -- 1) suitable independent combinations of the activities.
Eventually, we briefly argue how the activity series for the pressure, combined
with the Debye-dressing prescription, can be used for deriving approximate
equations of state at moderate densities, which include the contributions of
recombined entities made with three or more particles.
