  The collapse and fragmentation of filamentary primordial gas clouds are
explored using 1D and 2D hydrodynamical simulations coupled with the
nonequilibrium processes of H2 formation. The simulations show that depending
upon the initial density,there are two occasions for the fragmentation of
primordial filaments. If a filament has relatively low initial density, the
radial contraction is slow due to less effective H2 cooling. This filament
tends to fragment into dense clumps before the central density reaches
$10^{8-9}$ cm$^{-3}$, where H2 cooling by three-body reactions is effective and
the fragment mass is more massive than some tens $M_\odot$. In contrast, if a
filament is initially dense, the more effective H2 cooling with the help of
three-body reactions allows the filament to contract up to $n\sim 10^{12}$
cm$^{-3}$. After the density reaches $n\sim 10^{12}$ cm$^{-3}$, the filament
becomes optically thick to H2 lines and the radial contraction subsequently
almost stops. At this final hydrostatic stage, the fragment mass is lowered
down to $\approx 1M_\odot$ because of the high density of the filament. The
dependence of the fragment mass upon the initial density could be translated
into the dependence on the local amplitude of random Gaussian density fields or
the epoch of the collapse of a parent cloud. Hence, it is predicted that the
initial mass function of Population III stars is likely to be bimodal with
peaks of $\approx 10^2 M_\odot$ and $\approx 1M_\odot$, where the relative
heights could be a function of the collapse epoch.
