Context : The stellar halo provides precious information about the Galaxy in its early stages of evolution because the most metal-poor ( oldest ) stars in the Milky Way are found there .

Aims : We study the chemical evolution and formation of the Galactic halo through the analysis of its stellar metallicity distribution function and some key elemental abundance patterns .
We also test the effects of a possible population III of zero-metal stars .

Methods : Starting from the two-infall model for the Galaxy , which predicts too few low-metallicity stars , we add a gas outflow during the halo phase with a rate proportional to the star formation rate through a free parameter , \lambda .
In addition , we consider a first generation of massive zero-metal stars in this two-infall + outflow model adopting two different top-heavy initial mass functions and specific population III yields .

Results : The metallicity distribution function of halo stars , as predicted by the two-infall + outflow model shows a good agreement with observations , when the parameter \lambda = 14 and the time scale for the first infall , out of which the halo formed , is not longer than 0.2 Gyr , a lower value than suggested previously .
Moreover , the abundance patterns [ X/Fe ] vs. [ Fe/H ] for C , N and \alpha -elements O , Mg , Si , S , Ca show a good agreement with the observational data , as in the case of the two-infall model without outflow .
If population III stars are included , under the assumption of different initial mass functions , the overall agreement of the predicted stellar metallicity distribution function with observational data is poorer than in the case of the two-infall + outflow model without population III .

Conclusions : We conclude that it is fundamental to include both a gas infall and outflow during the halo formation to explain the observed halo metallicity distribution function , in the framework of a model assuming that the stars in the inner halo formed mostly in situ .
Moreover , we find that it does not exist a satisfactory initial mass function for population III stars which reproduces the observed halo metallicity distribution function .
As a consequence , there is no need for a first generation of only massive stars to explain the evolution of the Galactic halo .