  We consider the hydrogen quantum plasma in the Saha regime, where it almost
reduces to a partially ionized atomic gas. We briefly review the construction
of systematic expansions of thermodynamical functions beyond Saha theory, which
describes an ideal mixture of ionized protons, ionized electrons and hydrogen
atoms in their ground-state. Thanks to the existence of rigorous results, we
first identify the simultaneous low-temperature and low-density limit in which
Saha theory becomes asymptotically exact. Then, we argue that the screened
cluster representation is well suited for calculating corrections, since that
formalism accounts for all screening and recombination phenomena at work in a
more tractable way than other many-body methods. We sketch the corresponding
diagrammatical analysis, which leads to an exact asymptotic expansion for the
equation of state. That scaled low-temperature expansion improves the
analytical knowledge of the phase diagram. It also provides reliable numerical
values over a rather wide range of temperatures and densities, as confirmed by
comparisons to quantum Monte Carlo data.
