  We present analytical calculations and numerical simulations for the
synchronization of oscillators interacting via a long range power law
interaction on a one dimensional lattice. We have identified the critical value
of the power law exponent $\alpha_c$ across which a transition from a
synchronized to an unsynchronized state takes place for a sufficiently strong
but finite coupling strength in the large system limit. We find $\alpha_c=3/2$.
Frequency entrainment and phase ordering are discussed as a function of $\alpha
\geq 1$. The calculations are performed using an expansion about the aligned
phase state (spin-wave approximation) and a coarse graining approach. We also
generalize the spin-wave results to the {\it d}-dimensional problem.
