  The gauged $U(1)_{L_{\mu}-L_{\tau}}$ symmetry is the simplest possibility to
explain the observed muon $g-2$, while being consistent with the neutrino
oscillations through the seesaw mechanism. In this paper, we investigate if
leptogenesis can work at the same time. At first glance, leptogenesis seems
challenging because the right-handed neutrino masses are related to the
$U(1)_{L_{\mu}-L_{\tau}}$ breaking scale of $10\,$-$\,100\,$GeV as required
from the muon $g-2$. Contrary to this expectation, we find that non-thermal
leptogenesis with the right-handed neutrino masses of
$\mathcal{O}(10^{7})\,$GeV is possible. The successful scenario results in
strict predictions on the neutrino oscillation parameters, which will be tested
in future experiments.
