  We propose a covariant definition of the fractional Klein-Gordon equation
with long-range interactions independent of the metric of the underlying
manifold. As an example we consider the fractional Klein-Gordon equation on
AdS$_{2+1}$, computing the explicit kernel representation of the fractional
Laplace-Beltrami operator as well as the two-point propagator of the fractional
Klein-Gordon equation. Our results suggest that the propagator only exists if
the mass is small compared to the inverse AdS radius, presumably because the
AdS space expands faster with distance as a flat space of the same dimension.
Our results are expected to be useful in particular for new applications of the
AdS/CFT correspondence within statistical mechanics and quantum information.
