The dynamical Mertens' Theorem gives asymptotics for weighted averages of numbers of closed orbits in S-integer dynamical systems, which are constructed from arithmetic data, namely K = Q, ξ = 2, and S a subset of rational primes. The way of finding such asymptotic expressions is an analogue of Mertens' Theorem in analytic number theory. In this article, we focus on the dynamical Mertens' Theorem of some certain growths in case S and its complement are infinite subsets of all rational primes. More precisely, our aim is to find out a leading coefficient which comes from the main term in the asymptotic expression of our interested setting. Moreover, the real interval covering such a coefficient will be provided in some certain examples.
