  Cosmic metallicity evolution possibly creates the diversity of star formation
modes at different epochs. Gravitational fragmentation of circumstellar discs
provides an important formation channel of multiple star systems, including
close binaries. We here study the nature of disc fragmentation, systematically
performing a suite of two-dimensional radiation-hydrodynamic simulations, in a
broad range of metallicities, from the primordial to the solar values. In
particular, we follow relatively long-term disc evolution over 15 kyr after the
disc formation, incorporating the effect of heating by the protostellar
irradiation. Our results show that the disc fragmentation occurs at all
metallicities $1$--$0$ $Z_{\odot}$, yielding self-gravitating clumps. Physical
properties of the clumps, such as their number and mass distributions, change
with the metallicity due to different gas thermal evolution. For instance, the
number of clumps is the largest for the intermediate metallicity range of
$10^{-2}$--$10^{-5}$ $Z_{\odot}$, where the dust cooling is effective
exclusively in a dense part of the disc and causes the fragmentation of spiral
arms. The disc fragmentation is more modest for $1$--$0.1$ $Z_{\odot}$ thanks
to the disc stabilization by the stellar irradiation. Such metallicity
dependence agrees with the observed trend that the close binary fraction
increases with decreasing metallicity in the range of $1$--$10^{-3}$
$Z_{\odot}$.
