Functions concerned with divisors of order $r$

Abstract

N. Minculete has introduced a concept of divisors of order $r$: integer $d=p_1^{b_1}\cdots p_k^{b_k} $ is called a divisor of order $r$ of $n=p_1^{a_1}\cdots p_k^{a_k}$ if $d \mid n$ and $b_j\in\{r, a_j\}$ for $j=1,\ldots,k$. One can consider respective divisor function $\tau^{(r)}$ and sum-of-divisors function $\sigma^{(r)}$. In the present paper we investigate the asymptotic behaviour of $\sum_{n\le x} \tau^{(r)}(n)$ and $\sum_{n\le x} \sigma^{(r)}(n)$. We also provide conditional estimates under Riemann hypothesis.