  The Pristine survey uses narrow-band photometry to derive precise
metallicities down to the extremely metal-poor regime ([Fe/H] < -3), and
currently consists of over 4 million FGK-type stars over a sky area of $\sim
2~500\, \mathrm{deg}^2$. We focus our analysis on a subsample of $\sim 80~000$
main sequence turnoff stars with heliocentric distances between 6 and 20 kpc,
which we take to be a representative sample of the inner halo. The resulting
metallicity distribution function (MDF) has a peak at [Fe/H] = -1.6, and a
slope of $\Delta$(LogN)/$\Delta[Fe/H] = 1.0 \pm 0.1$ in the metallicity range
of -3.4 < [Fe/H] < -2.5. This agrees well with a simple closed-box chemical
enrichment model in this range, but is shallower than previous spectroscopic
MDFs presented in the literature, suggesting that there may be a larger
proportion of metal-poor stars in the inner halo than previously reported. We
identify the Monoceros/TriAnd/ACS/EBS/A13 structure in metallicity space in a
low latitude field in the anticenter direction, and also discuss the
possibility that the inner halo is dominated by a single, large merger event,
but cannot strongly support or refute this idea with the current data. Finally,
based on the MDF of field stars, we estimate the number of expected metal-poor
globular clusters in the Milky Way halo to be 5.4 for [Fe/H] < -2.5 and 1.5 for
[Fe/H] < -3, suggesting that the lack of low metallicity globular clusters in
the Milky Way is not due simply to statistical undersampling.
