Asymptotic formulae for certain arithmetic functions produced by fractional linear transformations

K.T. Atanassov introduced the two arithmetic functions \[ I(n) = \prod_{\nu=1}^k p_\nu^{1/\alpha_\nu} \qquad \text{and}\qquad R(n) = \prod_{\nu=1}^k p_\nu^{\alpha_v - 1} \] called the irrational factor and the strong restrictive factor, respectively. A variety of authors have studied the properties of these arithmetic functions. We consider weighted combinations $I(n)^\alpha R(n)^\beta$ and characterize pairs $(\alpha,\beta)$ in order to measure how close $n$ is to being $k$-power full or $k$-power free. We then generalize these functions to a class of arithmetic functions defined in terms of fractional linear transformations arising from certain $2 \times 2$ matrices, establish asymptotic formulae for averages of these functions, and explore certain maps that arise from considering the leading terms of these averages. We further generalize to a larger class of maps by introducing real moments, which allow us to explore new properties of these arithmetic functions. We additionally study the influence of the eigenvalues of a matrix on the associated arithmetic function, and obtain results on the local density of eigenvalues through their connection to a particular surface. Finally, we present a further generalization involving arithmetic functions defined by certain complex-valued fractional linear transformations, explore some of the properties of these new functions, and present a few open problems