Atanassov introduced the irrational factor function and the strong restrictive factor function, which he defined as $I(n)=displaystyleprod_{p^{alpha}||n}^{}p^{1/{alpha}}$ and $R(n)=displaystyleprod_{p^{alpha}||n}^{}p^{{alpha}-1}$ in [2] and [3]. Various properties of these functions have been investigated by Alkan, Ledoan, Panaitopol, and the authors. In the prequel, we expanded these functions to a class of elements of $PSL_2(mathbb{Z})$, and studied some of the properties of these maps. In the present paper we generalize the previous work by introducing real moments and considering a larger class of maps. This allows us to explore new properties of these arithmetic functions.