Functions concerned with divisors of order $r$

TL;DR

This paper studies the asymptotic behavior of divisor functions of order r, extending classical divisor functions, and provides estimates including those conditional on the Riemann hypothesis.

Contribution

It introduces and analyzes the asymptotic properties of new divisor functions of order r, extending classical divisor function theory.

Findings

Asymptotic formulas for sums of τ^{(r)}(n) and σ^{(r)}(n) up to x.

Conditional estimates under the Riemann hypothesis.

Extension of divisor function theory to 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.