  We determine the metallicity distribution function (MDF) of the Galactic halo
by means of a sample of 1638 metal-poor stars selected from the Hamburg/ESO
objective-prism survey (HES). The sample was corrected for minor biases
introduced by the strategy for spectroscopic follow-up observations of the
metal-poor candidates, namely "best and brightest stars first". [...] We
determined the selection function of the HES, which must be taken into account
for a proper comparison between the HES MDF with MDFs of other stellar
populations or those predicted by models of Galactic chemical evolution. The
latter show a reasonable agreement with the overall shape of the HES MDF for
[Fe/H] > -3.6, but only a model of Salvadori et al. (2007) with a critical
metallicity for low-mass star formation of Z_cr = 10^{-3.4} * Z_Sun reproduces
the sharp drop at [Fe/H] ~-3.6 present in the HES MDF. [...] A comparison of
the MDF of Galactic globular clusters and of dSph satellites to the Galaxy
shows qualitative agreement with the halo MDF, derived from the HES, once the
selection function of the latter is included. However, statistical tests show
that the differences between these are still highly significant. [ABSTRACT
ABRIDGED]
