  We discuss analogues of the prime number theorem for a hyperbolic rational
map f of degree at least two on the Riemann sphere. More precisely, we provide
counting estimates for the number of primitive periodic orbits of f ordered by
their multiplier, and also obtain equidistribution of the associated
holonomies; both estimates have power savings error terms. Our counting and
equidistribution results will follow from a study of dynamical zeta functions
that have been twisted by characters of $S^1$. We will show that these zeta
functions are non-vanishing on a half plane $\Re(s) > \delta - \epsilon$, where
$\delta$ is the Hausdorff dimension of the Julia set of f.
