  An analog of the prime number theorem for a class of branched covering maps
on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological
models of some non-uniformly expanding rational maps without any smoothness or
holomorphicity assumption, is obtained in the second paper of this series. More
precisely, the number of primitive periodic orbits, ordered by a weight on each
point induced by a non-constant (eventually) positive real-valued H\"{o}lder
continuous function on $S^2$ satisfying the $\alpha$-strong non-integrability
condition, is asymptotically the same as the well-known logarithmic integral,
with an exponential error term. In this third and last paper of the series, we
show that the $\alpha$-strong non-integrability condition is generic in the
class of $\alpha$-H\"{o}lder continuous functions.
