  Previous theoretical studies suggest that the Population III (Pop3) stars
tend to form in extremely metal poor gas clouds with approximately $10^5
M_\odot$ embedded in mini dark matter halos. Very massive stars can form via
multiple collisions in Pop3 star clusters and eventually evolve to
intermediate-mass black holes (IMBHs). In this work, we conduct star-by-star
$N$-body simulations for modelling the long-term evolution of Pop3 star
clusters. We find that if the mini dark matter halos can survive today, these
star clusters can avoid tidal disruption by the galactic environment and can
efficiently produce IMBH-BH mergers among a wide range of redshift from 0 to
20. The average gravitational wave event rate is estimated to be
$0.1-0.8~\mathrm{yr}^{-1} \mathrm{Gpc}^{-3}$, and approximately $40-80$ percent
of the mergers occur at high redshift ($z>6$). The characteristic strain shows
that a part of low-redshift mergers can be detected by LISA, TianQin, and
Taiji, whereas most mergers can be covered by DECIGO and advanced
LIGO/VIRGO/Kagra. Mergers with pair-instability BHs have a rate of
approximately $0.01-0.15$~yr$^{-1}$~Gpc$^{-3}$, which can explain the
GW190521-like events.
