The Pristine Survey - VIII. The metallicity distribution function of the Milky Way halo down to the extremely metal-poor regime

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 similar to 2500 deg(2), We focus our analysis on a subsample of similar to 80 000 main-sequence turn-off 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 +/- 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, hut 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/EIRS/A13 structure in metallicity space in a low-latitude field in the anticentre 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 he 5,4 for [Fe/II] <-2.5 and 1.5 for [Fe/II] <-3, suggesting that the lack of lowmetallicity globular clusters in the Milky Way is not due simply to statistical undersampling.