  Because massive, low-metallicity population III (PopIII) stars may produce
very powerful long gamma-ray bursts (LGRBs), high-redshift GRB observations
could probe the properties of the first stars. We analyze the correlation
between early PopIII stars and LGRBs by using cosmological
N-body/hydrodynamical simulations, which include detailed chemical evolution,
cooling, star formation, feedback effects and the transition between PopIII and
more standard population I/II (PopII/I) stars. From the Swift observed rate of
LGRBs, we estimate the fraction of black holes that will produce a GRB from
PopII/I stars to be in the range 0.028<f_{GRB}<0.140, depending on the assumed
upper metallicity of the progenitor. Assuming that as of today no GRB event has
been associated to a PopIII star, we estimate the upper limit for the fraction
of LGRBs produced by PopIII stars to be in the range 0.006<f_{GRB}<0.022. When
we apply a detection threshold compatible with the BAT instrument, we find that
the expected fraction of PopIII GRBs (GRB3) is ~10% of the full LGRB population
at z>6, becoming as high has 40% at z>10. Finally, we study the properties of
the galaxies hosting our sample of GRB3. We find that the average metallicity
of the galaxies hosting a GRB3 is typically higher than the critical
metallicity used to select the PopIII stars, due to the efficiency in polluting
the gas above such low values. We also find that the highest probability of
finding a GRB3 is within galaxies with a stellar mass <10^7 Msun, independently
from the redshift.
