An order result for the exponential divisor function

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].