  Recent evidence appears to link gamma-ray bursts (GRBs) to star-forming
regions in galaxies at cosmological distances. If short-lived massive stars are
the progenitors of GRBs, the rate of events per unit cosmological volume should
be an unbiased tracer (i.e. unaffected by dust obscuration and surface
brightness limits) of the cosmic history of star formation. Here we use
realistic estimates for the evolution of the stellar birthrate in galaxies to
model the number counts, redshift distribution, and time-delay factors of GRBs.
We present luminosity function fits to the BATSE log N-log P relation for
different redshift distributions of the bursts. Our results imply about 1-2
GRBs every one million Type II supernovae, and a characteristic
`isotropic-equivalent' burst luminosity in the range 3-20 x 1e51 ergs/s (for
H_0=65 km/s/Mpc). We compute the rate of multiple imaging of background GRBs
due to foreground mass condensations in a Lambda-dominated cold dark matter
cosmology, assuming that dark halos approximate singular isothermal spheres on
galaxy scales and Navarro-Frenk-White profiles on group/cluster scales, and are
distributed in mass according to the Press-Schechter model. We show that the
expected sensivity increase of Swift relative to BATSE could result in a few
strongly lensed individual bursts detected down to a photon flux of 0.1
phot/s/cm^2 in a 3-year survey. Because of the partial sky coverage, however,
it is unlikely that the Swift satellite will observe recurrent events (lensed
pairs).
