Constraints on the abundance of primordial black holes with different
  mass distributions from lensing of fast radio bursts
  Primordial black holes (PBHs) has been considered to form a part of dark
matter for a long time but the possibility has been poorly constrained over a
wide mass range, including the stellar mass range ($1-100~M_{\odot}$). However,
due to the discovery of merger events of black hole binaries by LIGO-Virgo
gravitational wave observatories, the interest for PBHs in the stellar mass
window has been aroused again. Fast radio bursts (FRBs) are bright radio
transients with millisecond duration and very high all-sky occurrence rate.
Lensing effect of these bursts has been proposed as one of the optimal probes
for constraining the abundance of PBHs in the stellar mass range. In this
paper, we first investigate constraints on the abundance of PBHs from the
latest $593$ FRB observations for both the monochromatic mass distribution and
three other popular extended mass distributions related to different formation
mechanisms of PBHs. It is found that constraints from currently public FRB
observations are relatively weaker than those from existing gravitational wave
detections. Furthermore, we forecast constraining power of future FRB
observations on the abundance of PBHs with different mass distributions of PBHs
and different redshift distributions of FRBs taken into account. Finally, We
find that constraints of parameter space on extended mass distributions from
$\sim10^5$ FRBs with $\overline{\Delta t}\leq1 ~\rm ms$ would be comparable
with what can be constrained from gravitational wave events. It is foreseen
that upcoming complementary multi-messenger observations will yield
considerable constraints on the possibilities of PBHs in this intriguing mass
window.
