  We obtain an analogue of the prime number theorem for a class of branched
covering maps on the $2$-sphere called expanding Thurston maps $f$, which are
topological models of some rational maps without any smoothness or
holomorphicity assumption. More precisely, by studying dynamical zeta functions
and, more generally, dynamical Dirichlet series for $f$, we show that the
number of primitive periodic orbits of $f$, ordered by a weight on each point
induced by a non-constant (eventually) positive real-valued H\"{o}lder
continuous function $\phi$ on $S^2$ satisfying some additional regularity
conditions, is asymptotically the same as the well-known logarithmic integral,
with an exponential error term. Such a result, known as a Prime Orbit Theorem,
follows from our quantitative study of the holomorphic extension properties of
the associated dynamical zeta functions and dynamical Dirichlet series. In
particular, the above result applies to postcritically-finite rational maps
whose Julia set is the whole Riemann sphere. Moreover, we prove that the
regularity conditions needed here are generic; and for a Latt\`{e}s map $f$ and
a continuously differentiable (eventually) positive function $\phi$, such a
Prime Orbit Theorem holds if and only if $\phi$ is not co-homologous to a
constant.
