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. 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. 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. 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 without population III. 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.

Accepted for publication in A&A. 11 pages, 5 figures