  Population (Pop) III stars, first stars, or metal-free stars are made of
primordial gas. We have examined if they can be dominant origins of merging
binary black holes (BHs) and extremely metal-poor stars. The abundance pattern
of EMP stars is helpful to trace back the properties of Pop III stars. We have
confirmed previous arguments that the observed BH merger rate needs Pop III
star formation efficiency 10 times larger than theoretically predicted values,
while the cosmic reionization history still permits such a high Pop III star
formation efficiency. On the other hand, we have newly found that the elemental
abundance pattern of EMP stars only allows the Pop III initial mass function
with the minimum mass of $\sim 15 - 27$ $M_\odot$. In other words, the minimum
mass must not deviate largely from the critical mass below and above which Pop
III stars leave behind neutron stars and BHs, respectively. Pop III stars may
be still a dominant origin of merging binary BHs but our study has reduced the
allowed parameter space under a hypothesis that EMP stars are formed from
primordial gas mixed with Pop III supernova ejecta.
