  Despite the great attention devoted to the study of phase oscillators on
complex networks in the last two decades, it remains unclear whether scale-free
networks exhibit a nonzero critical coupling strength for the onset of
synchronization in the thermodynamic limit. Here, we systematically compare
predictions from the heterogeneous degree mean-field (HMF) and the quenched
mean-field (QMF) approaches to extensive numerical simulations on large
networks. We provide compelling evidence that the critical coupling vanishes as
the number of oscillators increases for scale-free networks characterized by a
power-law degree distribution with an exponent $2 < \gamma \leq 3$, in line
with what has been observed for other dynamical processes in such networks. For
$\gamma > 3$, we show that the critical coupling remains finite, in agreement
with HMF calculations and highlight phenomenological differences between
critical properties of phase oscillators and epidemic models on scale-free
networks. Finally, we also discuss at length a key choice when studying
synchronization phenomena in complex networks, namely, how to normalize the
coupling between oscillators.
