Exponential divisor functions

Consider the operator $E$ on arithmetic functions such that $Ef$ is the multiplicative arithmetic function defined by $(Ef)(p^a)=f(a)$ for every prime power $p^a$. We investigate the behaviour of $E^m{\tau }_k$, where ${\tau }_k$ is a $k$-dimensional divisor function and $E^m$ stands for the $m$-fold iterate of $E$. We estimate the error terms of ${\sum }_{n\le x}{E}^m{\tau }_k(n)$ for various combinations of $m$ and $k$. We also study properties of $E^{m}f$ for arbitrary $f$ and sufficiently large $m$. Our study provides a unified approach to functions with exponential divisors. We improve special cases of the Dirichlet asymmetric divisor problem and several results on the exponential divisor and totient functions.