  A sofic shift is a shift space consisting of bi-infinite labels of paths from
a labelled graph. Being a dynamical system, the distribution of its closed
orbits may indicate the complexity of the space. For this purpose, prime orbit
and Mertens' orbit counting functions are introduced as a way to describe the
growth of the closed orbits. The asymptotic behaviours of these counting
functions can be implied from the analiticity of the Artin-Mazur zeta function
of the space. Despite having a closed-form expression, the zeta function is
expressed implicitly in terms of several signed subset matrices, and this makes
the study on its analyticity to be seemingly difficult. In this paper, we will
prove the asymptotic behaviours of the counting functions for a sofic shift via
its zeta function. This involves investigating the properties of the said
matrices. Suprisingly, the proof is rather short and only uses well-known facts
about a sofic shift, especially on its minimal right-resolving presentation.
