An order result for the exponential divisor function

TL;DR

This paper derives an asymptotic formula with a remainder term for the r-th power of the exponential divisor counting function, improving previous results in the field of multiplicative number theory.

Contribution

It establishes a refined asymptotic formula for the r-th power of the exponential divisor function, advancing understanding of its growth and distribution.

Findings

Derived an asymptotic formula with a remainder term

Improved upon previous results by Subbarao

Enhanced understanding of exponential divisor function behavior

Abstract

The integer $d=\prod_{i=1}^s p_i^{b_i}$ is called an exponential divisor of $n=\prod_{i=1}^s p_i^{a_i}>1$ if $b_i \mid a_i$ for every $i\in \{1,2,...,s\}$. Let $\tau^{(e)}(n)$ denote the number of exponential divisors of $n$, where $\tau^{(e)}(1)=1$ by convention. The aim of the present paper is to establish an asymptotic formula with remainder term for the $r$-th power of the function $\tau^{(e)}$, where $r\ge 1$ is an integer. This improves an earlier result of {\sc M. V. Subbarao} [5].