Introduction {#sec:introduction}
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It is widely accepted that the so called Population III stars, i.e. the first stars formed from the primordial gas with an extremely low metallicity, have played an important role in the evolution of the universe. The first stars are assumed to be predominantly very massive and they were responsible for reionization of the universe and its enrichment by heavy elements.

The formation of the first stars and hence also their Initial Mass Function (IMF) is significantly influenced by the low metallicity of the protostellar material. Due to its low opacity, its gravitational attraction can not be effectively opposed by radiative pressure and the continuing accretion gives rise to more massive stars.

Numerical hydrodynamic simulations of star formation generally confirm the tendency to higher masses of the first stars, although quantitatively the results differ considerably.

[@Bromm2004] found the masses of the first stars between $10^{2}$ and $10^{3}M_{\odot}$, [@Omukai2003] followed the evolution of accreting protostars and found $\sim 600\,M_{\odot}$ as the upper limit for massive stars. According to [@Abel2002], the final masses are uncertain because a single molecular protostar seed of $\sim 1 M_{\odot}$ at the centre of a $\sim 100\,M_{\odot}$ core of a protogalaxy is formed. The 