  We use the moving-mesh code AREPO to investigate the effects of different
levels of rotation and turbulence on the fragmentation of primordial gas and
the formation of Population III stars. We consider 9 different combinations of
turbulence and rotation and carry out 5 different realizations of each setup,
yielding one of the largest sets of simulations of Population III star
formation ever performed. We find that fragmentation in Population III
star-forming systems is a highly chaotic process and show that the outcomes of
individual realizations of the same initial conditions often vary
significantly. However, some general trends are apparent. Increasing the
turbulent energy promotes fragmentation, while increasing the rotational energy
inhibits fragmentation. Within the 1000 yr period that we simulate, runs
including turbulence yield flat protostellar mass functions while purely
rotational runs show a more top-heavy distribution. The masses of the
individual protostars are distributed over a wide range from a few $10^{-3} \,
{\rm M_{\odot}}$ to several tens of ${\rm M_\odot}$. The total mass growth rate
of the stellar systems remains high throughout the simulations and depends only
weakly on the degree of rotation and turbulence. Mergers between protostars are
common, but predictions of the merger fraction are highly sensitive to the
criterion used to decide whether two protostars should merge. Previous studies
of Population III star formation have often considered only one realization per
set of initial conditions. However, our results demonstrate that robust trends
can only be reliably identified by considering averages over a larger sample of
runs.
