Exponential divisor functions

TL;DR

This paper studies the behavior of exponential divisor functions and their iterates, providing estimates for sums and properties, thereby unifying approaches to functions with exponential divisors and improving existing divisor problem results.

Contribution

It introduces a unified framework for analyzing exponential divisor functions and their iterates, improving bounds and understanding of related divisor and totient functions.

Findings

Estimated error terms for sums of iterated exponential divisor functions

Established properties of functions after multiple iterations of the operator E

Enhanced bounds for the Dirichlet asymmetric divisor problem

Abstract

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 $E^mf$ 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.