Mean Value of r-gcd-sum and r-lcm-Sum Functions

Abstract

In this paper we perform a further investigation for r-gcd-sum and r-lcm-sum functions. By making use of the properties of generalization of Euler’s $\phi$-function, Abel’s identity and elementary arguments, we derive asymptotic formulas for the average of the r-gcd-sum function, r-lcm-sum function and their generalizations. Moreover, we also study the sums of reciprocals of r-gcd and r-lcm.

1. Introduction

For any integer $n\ge 1$, the Pillai’s arithmetical function, which is also known as the gcd-sum function is defined by

$$P\left(n\right)=\sum _{k=1}^n(k,n)=\sum _{d|n}d\phi (n/d),$$

(1)

where $\phi$ is the Euler’s totient function, and $(k,n)$ denotes the greatest common divisor of k and n. In the past several decades, the study of the gcd-sum function arises naturally in number theory, and a large number of mathematicians have investigated its distribution, arithmetic and algebraic properties. For example, an asymptotic formula for the average order of $P\left(n\right)$ was proved by Bordellés [1], namely, for every $\epsilon >0$,

$$\sum _{n\le x}P\left(n\right)=\frac{x^2}{2\zeta \left(2\right)}\left(\log x+2\gamma -\frac{1}{2}-\frac{{\zeta }^{{}^{\prime }}\left(2\right)}{\zeta \left(2\right)}\right)+O\left(x^{1+\theta +\epsilon }\right),$$

(2)

where $\gamma$ is Euler’s constant and $\theta$ is the exponent appearing in the Dirichlet’s divisor problem. In addition, Bordellés [2] also studied the sum of reciprocal of the gcd and deduced that

$$\sum _{n\le x}{P}^{(-1)}\left(n\right)=\sum _{n\le x}\sum _{k=1}^n\frac{1}{(k,n)}=\frac{\zeta \left(3\right)}{2\zeta \left(2\right)}{x}^2+O\left(x{\left(\log x\right)}^{2/3}{\left(\log \log x\right)}^{4/3}\right).$$

(3)

On the other hand, let $[k,n]$ denote the least common multiple of k and n. The lcm-sum function is defined as

$$L\left(n\right)=\sum _{k=1}^n[k,n]=\frac{n}{2}\left(1+\sum _{d|n}d\phi (n/d)\right)$$

(4)

for any integer $n\ge 1$. Bordellés investigated the function $L\left(n\right)$ in his article [2] and established that

$$\sum _{n\le x}L\left(n\right)=\frac{\zeta \left(3\right)}{8\zeta \left(2\right)}{x}^4+O\left(x^3{\left(\log x\right)}^{2/3}{\left(\log \log x\right)}^{4/3}\right),$$

(5)

$$\sum _{n\le x}{L}^{(-1)}\left(n\right)=\sum _{n\le x}\sum _{k=1}^n\frac{1}{[k,n]}=\frac{{\left(\log x\right)}^3}{6\zeta \left(2\right)}+\frac{{\left(\log x\right)}^2}{2\zeta \left(2\right)}\left(\gamma +\log \left(\frac{{\mathcal{A}}^{12}}{2\pi }\right)\right)+O\left(\log x\right),$$

(6)

where $\mathcal{A}\approx 1.282427\cdots$ is the Glaisher–Kinkelin constant.

Let $r\ge 1$ be a fixed integer, the greatest $r^{th}$ power common divisor of positive integers a and b is defined to be the largest positive integer $d^r$ such that $d^r|a$ and $d^r|b$, which is denoted by ${(a,b)}_r$ and called the r-gcd of a and b. Note that ${(a,b)}_1=(a,b)$. The related research on greatest $r^{\mathrm{th}}$ power common divisor can be found in [3]. Furthermore, we know that the most natural geometric usage of the gcd function is in counting visible lattice points. Recently, a new notion of visibility has gained much attention, where curves replace straight lines as the lines of sight. For example [4], A point $(k,n)\in {\mathbb{Z}}^2$ is visible along the the curve $y=h{x}^r$ if and only if $max\{d\ge 1:d|kand{d}^r\left|n\right\}=1$, where $r\in \mathbb{N}$ and $h\in \mathbb{Q}$. So the immediate natural extension is to consider visibility along curves $y=h{x}^r$ or $x=h{y}^r$. It is obvious that the r-gcd function provides a necessary criterion for the combined visibility of a lattice point along the curves $y=h{x}^r$ or $x=h{y}^r$. So the fundamental research on r-gcd function is important in future work.

Currently, Prasad, Reddy and Rao [5] have introduced a natural generalization of the usual gcd-sum function as follows:

$$P_r\left(n^r\right)=\sum _{k=1}^{n^r}{(k,n^r)}_r,$$

(7)

which is called r-gcd-sum function. They obtained an asymptotic formula for its summatory function:

$$\sum _{n^r\le x}{P}_r\left(n^r\right)=\frac{x^{1+\frac{1}{r}}}{(r+1)\zeta (r+1)}\left(\frac{\log x}{r}+2\gamma -\frac{1}{r+1}-\frac{{\zeta }^{{}^{\prime }}(r+1)}{\zeta (r+1)}\right)+O_{\epsilon }\left(x^{1+\frac{\theta +\epsilon }{r}}\right).$$

(8)

With the help of (8), the authors [5] studied the partial sums of the Dirichlet series of the r-gcd-sum function, that is

$$G_{t,r}\left(x\right)\sum _{n^r\le x}\frac{P_r\left(n^r\right)}{n^t},$$

for all real values of t and any integer $r\ge 1$, and established asymptotic formulas for this function. Moreover, they [6] also introduced an r-gcd-sum function over r-regular integers $\left(\mathrm{mod}{n}^r\right)$, which is defined by

$$\widetilde{P_r}\left(n^r\right):=\sum _{kϵRe{g}_r\left(n^r\right)}{(k,n^r)}_r,$$

where $Re{g}_r\left(n^r\right)$ = {$k:1\le k\le n^r$, k is r-regular mod $n^r$}. Based on the properties of a generalization of Euler’s $\phi$-function [3], they obtained some arithmetic properties of $\widetilde{P_r}\left(n^r\right)$ and an asymptotic formula for its summatory function.

On the other hand, it is natural to consider the generalized least common multiple related to r-gcd. In their article [7], Bu and Xu defined the least $r^{\mathit{th}}$ power common multiple as

$${[k,n]}_r=\frac{kn}{{(k,n)}_r},$$

which is also called r-lcm of k and n. They proved that

$$\sum _{n\le x}\sum _{k=1}^n{\left({[k,n]}_r\right)}^a=\frac{x^{2a+2}\zeta (ar+2r)}{2{(a+1)}^2\zeta \left(2r\right)}+O\left(x^{2a+1}\right)$$

(9)

with positive integer $a>1$.

We notice that the r-gcd appeared when studying the r-free integers, and it also has most commonly been used to study the generalized totient function or commonly called the Klee’s totient function. Moreover, because the function $P_r\left(n^r\right)$ is multiplicative, it is an interesting object of study from an analytic point of view, and the question is also significant to investigate the distribution of r-gcd and r-lcm by estimating the average of r-gcd-sum function and r-lcm-sum function.

The purpose of this paper is to perform a further investigation for r-gcd-sum and r-lcm-sum functions. By making use of the properties of a generalization of Euler’s $\phi$-function, Abel’s identity and elementary arguments, we derive asymptotic formulas for the average of the r-gcd-sum function, r-lcm-sum function and their generalizations. Furthermore, we also study the sums of reciprocals of r-gcd and r-lcm.

2. Main Results

In this section, we give the main results of this paper as follows.

Theorem 1.

For any real number $x>1$ and integer $r>1$, we have

$$\sum _{n\le x}\sum _{k=1}^n{(k,n)}_r=\frac{\zeta \left(r\right)}{2\zeta \left(2r\right)}{x}^2+O\left(x^{1+\frac{1}{r}}\log x\right).$$

(10)

Notice that for $r=1$, the asymptotic formula for average of the usual gcd-sum function $P\left(n\right)$ had been done in [1] by the convolution identity $P=\phi \ast Id=\mu \ast (Id·\tau )$, where $Id\left(n\right)=n$ and ∗ is the usual Dirichlet convolution product.

Theorem 2.

For any real number $x>1$, we have the following estimate

$$\sum _{n\le x}\sum _{k=1}^n\frac{1}{{(k,n)}_r}=\frac{\zeta \left(3r\right)}{2\zeta \left(2r\right)}{x}^2+O\left(x\log x\right).$$

(11)

When $r=1$, the main term of estimate (11) is equal to the main term of (3). Moreover, the error term of estimate (3) is improved from $O\left(x\log x\right)$ to $O\left(x{\left(\log x\right)}^{2/3}{\left(\log \log x\right)}^{4/3}\right)$ because of the classical result of Walfisz [8],

$$R\left(x\right):=\sum _{n\le x}\phi \left(n\right)-\frac{1}{2\zeta \left(2\right)}{x}^2=O\left(x{\left(\log x\right)}^{2/3}{\left(\log \log x\right)}^{4/3}\right).$$

Theorem 3.

Let $a\ge 2$ be a positive integer, then we obtain

$$\sum _{n\le x}\sum _{k=1}^n{\left({(k,n)}_r\right)}^a=\frac{\zeta \left(a\right)}{(a+1)\zeta (ra+r)}{x}^{a+1}+O\left(x^a\right).$$

(12)

Theorem 4.

Let a be a positive integer, then for any real number $x>e$ sufficiently large, we have the following estimate

$$\sum _{n\le x}\sum _{k=1}^n{\left({[k,n]}_r\right)}^a=\frac{\zeta (ar+2r)}{2{(a+1)}^2\zeta \left(2r\right)}{x}^{2a+2}+O\left(x^{2a+1}\log x\right).$$

(13)

Here we offer a more simple proof of ${\sum }_{n\le x}{\sum }_{k=1}^n{\left({(k,n)}_r\right)}^a$ by using elementary calculations.

Taking $a=1$ in Theorem 4, we can get

Corollary 1.

For any real number $x>e$ sufficiently large,

$$\sum _{n\le x}\sum _{k=1}^n{[k,n]}_r=\frac{\zeta \left(3r\right)}{8\zeta \left(2r\right)}{x}^4+O\left(x^3\log x\right).$$

(14)

Theorem 5.

For any real number $x>1$, we have

$$\sum _{n\le x}\sum _{k=1}^n\frac{1}{{[k,n]}_r}=\frac{{\left(\log x\right)}^3}{6\zeta \left(2r\right)}+A{\left(\log x\right)}^2+O\left(\log x\right),$$

(15)

where$A=\frac{\gamma }{\zeta \left(2r\right)}-\frac{{\zeta }^{{}^{\prime }}\left(2r\right)}{{\zeta }^2\left(2r\right)}$.

We know that for any function $f:\mathbb{N}\to \mathbb{C}$, there is the general identity

$$\sum _{m,n\le x}f(m,n)=2\sum _{n\le x}\sum _{m=1}^{n}f(m,n)-\sum _{n\le x}f(n,n),$$

which is symmetric in the variables. So it is easy to get the following asymptotic formulas from (10), (11), (14) and (15).

Corollary 2.

$$\sum _{m,n\le x}{(m,n)}_r=\frac{\zeta \left(r\right)}{\zeta \left(2r\right)}{x}^2+O\left(x^{1+\frac{1}{r}}\log x\right),$$

$$\sum _{m,n\le x}\frac{1}{{(m,n)}_r}=\frac{\zeta \left(3r\right)}{\zeta \left(2r\right)}{x}^2+O\left(x\log x\right),$$

$$\sum _{m,n\le x}{[m,n]}_r=\frac{\zeta \left(3r\right)}{4\zeta \left(2r\right)}{x}^4+O\left(x^3\log x\right),$$

$$\sum _{m,n\le x}\frac{1}{{[m,n]}_r}=\frac{{\left(\log x\right)}^3}{3\zeta \left(2r\right)}+2A{\left(\log x\right)}^2+O\left(\log x\right).$$

3. Proofs of Theorems

3.1. Proof of Theorems 1 and 2

Proof of Theorem 1. 

We know that an integer is rth-power-free if it is not divisible by the rth power of any integer $>1$. Then let ${\varphi }_r\left(n\right)$ denote the number of integers k in the set $1,\cdots ,n$, for which the greatest common divisor $(k,n)$ is rth-power-free, and ${\varphi }_r\left(n\right)$ is the generalization of the Euler’s $\phi$ function attributed to Klee [3]. The function ${\mu }^r\left(n\right)$ is defined as follows:

$${\mu }^r\left(1\right)=1;$$

if $n>1$ and $n=p_1^{a_1}{p}_2^{a_2}\cdots p_t^{a_t}$ is the canonical factorization of n, then

$$\begin{array}{c}\hfill {\mu }^r\left(n\right)=\left\{\begin{array}{cc}{(-1)}^t,\hfill & \mathrm{if}{a}_1=a_2=\cdots =a_t=r,\hfill \\ 0,\hfill & \mathrm{if}\mathrm{for}\mathrm{some}i,a_i\ne r.\hfill \end{array}\right.\end{array}$$

There is the following relationship between the function ${\varphi }_r\left(n\right)$ and function ${\mu }^r\left(n\right)$:

$${\varphi }_r\left(n\right)=n\sum _{d|n}{\mu }^r\left(d\right)/d.$$

So we can deduce

$$\begin{array}{cc}\hfill \sum _{n\le x}{\varphi }_r\left(n\right)=& \sum _{n\le x}\sum _{de=n}{\mu }^r\left(d\right)e=\sum _{d\le x}{\mu }^r\left(d\right)\sum _{e\le x/d}e\hfill \\ \hfill =& \frac{1}{2}{x}^2\sum _{d\le x}\frac{{\mu }^r\left(d\right)}{d^2}+O\left(x\sum _{d\le x}\frac{1}{d}\right)\hfill \\ \hfill =& \frac{1}{2}{x}^2\sum _{d=1}^{\infty }\frac{{\mu }^r\left(d\right)}{d^2}+\left(x^2\sum _{d>x}\frac{{\mu }^r\left(d\right)}{d^2}\right)+O\left(x\log x\right)\hfill \\ \hfill =& \frac{x^2}{2\zeta \left(2r\right)}+O\left(x\log x\right).\hfill \end{array}$$

(16)

It is easily seen from the definition of function ${\varphi }_r\left(n\right)$ that

$$\begin{array}{cc}\hfill \sum _{k=1}^n{(k,n)}_r=& \underset{{(k,n)}_r=d^r}{\sum _{k=1}^n}{d}^r=\sum _{d^r|n}{d}^r\underset{{(k,n/d^r)}_r=1}{\sum _{k=1}^{n/d^r}}1=\sum _{d^r|n}{d}^r{\varphi }_r\left(\frac{n}{d^r}\right).\hfill \end{array}$$

(17)

Hence, by applying (16) to (17), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _{n\le x}\sum _{k=1}^n{(k,n)}_r=& \sum _{n\le x}\sum _{d^r|n}{d}^r{\varphi }_r\left(\frac{n}{d^r}\right)=\sum _{d^r\le x}{d}^r\sum _{e\le x/d^r}{\varphi }_r\left(e\right)\hfill \\ \hfill =& \sum _{d^r\le x}{d}^r\left(\frac{1}{2\zeta \left(2r\right)}{\left(\frac{x}{d^r}\right)}^2+O\left(\frac{x}{d^r}\log \frac{x}{d^r}\right)\right)\hfill \\ \hfill =& \frac{x^2}{2\zeta \left(2r\right)}\sum _{d=1}^{\infty }\frac{1}{d^r}+\left(x^2\sum _{d>x^{1/r}}\frac{1}{d^r}\right)+O\left(x\log x\sum _{d^r\le x}1\right)\hfill \\ \hfill =& \frac{\zeta \left(r\right)}{2\zeta \left(2r\right)}{x}^2+O\left(x^{1+\frac{1}{r}}\log x\right).\hfill \end{array}\end{array}$$

This completes the proof of Theorem 1. □

The proof of Theorem 2 is similar to Theorem 1 by applying (16) easily, so we will not go into details here.

3.2. Proof of Theorem 3

Proof of Theorem 3. 

For $a\ge 0$, there is the elementary asymptotic formula

$$\sum _{n\le x}{n}^a=\frac{x^{a+1}}{a+1}+O\left(x^a\right),$$

then we have

$$\begin{array}{cc}\hfill \sum _{n\le x}\sum _{k=1}^n{\left({(k,n)}_r\right)}^a=& \sum _{n\le x}\sum _{d^r|n}{d}^{ra}{\varphi }_r\left(\frac{n}{d^r}\right)=\sum _{e\le x}{\varphi }_r\left(e\right)\sum _{d^r\le x/e}{d}^{ra}\hfill \\ \hfill =& \sum _{e\le x}{\varphi }_r\left(e\right)\left(\frac{1}{a+1}{\left(\frac{x}{e}\right)}^{a+1}+O\left({\left(\frac{x}{e}\right)}^a\right)\right)\hfill \\ \hfill =& \frac{x^{a+1}}{a+1}\sum _{e\le x}\frac{{\varphi }_r\left(e\right)}{e^{a+1}}+O\left(x^a\sum _{e\le x}\frac{{\varphi }_r\left(e\right)}{e^a}\right).\hfill \end{array}$$

(18)

Since ${\sum }_{n\le x}\frac{1}{n^s}=\frac{x^{1-s}}{1-s}+\zeta \left(s\right)+O\left(x^{-s}\right)$, if $s>0$ and $s\ne 1$, then we can deduce

$$\begin{array}{cc}\hfill \sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^a}=& \sum _{n\le x}\frac{1}{n^a}\sum _{d|n}{\mu }^r\left(d\right)\frac{n}{d}=\sum _{d\le x}\frac{{\mu }^r\left(d\right)}{d^a}\sum _{e\le x/d}\frac{1}{e^{a-1}}\hfill \\ \hfill =& \sum _{d\le x}\frac{{\mu }^r\left(d\right)}{d^a}\left(\frac{1}{2-a}{\left(\frac{x}{d}\right)}^{2-a}+\zeta (a-1)+O\left({\left(\frac{x}{d}\right)}^{1-a}\right)\right)\hfill \\ \hfill =& \frac{1}{2-a}{x}^{2-a}\sum _{d\le x}\frac{{\mu }^r\left(d\right)}{d^2}+\zeta (a-1)\sum _{d\le x}\frac{{\mu }^r\left(d\right)}{d^a}+O\left(x^{1-a}\sum _{d\le x}\frac{{\mu }^r\left(d\right)}{d}\right)\hfill \\ \hfill =& \frac{x^{2-a}}{(2-a)\zeta \left(2r\right)}+\left(x^{2-a}\sum _{d>x}\frac{{\mu }^r\left(d\right)}{d^2}\right)+\frac{\zeta (a-1)}{\zeta \left(ar\right)}+\left(\sum _{d>x}\frac{{\mu }^r\left(d\right)}{d^a}\right)\hfill \\ \hfill & +O\left(x^{1-a}\log x\right)\hfill \\ \hfill =& \frac{x^{2-a}}{(2-a)\zeta \left(2r\right)}+\frac{\zeta (a-1)}{\zeta \left(ar\right)}+O\left(x^{1-a}\log x\right),\hfill \end{array}$$

(19)

where $a>1$ and $a\ne 2$. So, it follows from (18) and (19) that

$$\begin{array}{cc}\hfill \sum _{n\le x}\sum _{k=1}^n{\left({(k,n)}_r\right)}^a=& \frac{x^{a+1}}{a+1}\left(\frac{x^{1-a}}{(1-a)\zeta \left(2r\right)}+\frac{\zeta \left(a\right)}{\zeta (ar+r)}+O\left(x^{-a}\log x\right)\right)\hfill \\ \hfill & +O\left(x^a\left(\frac{x^{2-a}}{(2-a)\zeta \left(2r\right)}+\frac{\zeta (a-1)}{\zeta \left(ar\right)}+O\left(x^{1-a}\log x\right)\right)\right)\hfill \\ \hfill =& \frac{\zeta \left(a\right)}{(a+1)\zeta (ra+r)}{x}^{a+1}+O\left(x^a\right),\hfill \end{array}$$

with $a\ge 2$.

This completes the proof of Theorem 3. □

3.3. Proof of Theorem 4

Proof of Theorem 4. 

From the definition of r-lcm-sum function, we have

$$\begin{array}{c}\hfill \sum _{k=1}^n{\left({[k,n]}_r\right)}^a=\sum _{k=1}^n{\left(\frac{kn}{{(k,n)}_r}\right)}^a=\sum _{d^r|n}\frac{n^a}{d^{ra}}\underset{{(k,n/d^r)}_r=1}{\sum _{k=1}^{n/d^r}}{\left(k{d}^r\right)}^a=n^a\sum _{d^r|n}\underset{{(k,n/d^r)}_r=1}{\sum _{k=1}^{n/d^r}}{k}^a,\end{array}$$

and for $s\ge 0$, notice that

$$\begin{array}{ccc}\hfill \underset{{(k,n)}_r=1}{\sum _{k=1}^n}{k}^s& =& \sum _{k=1}^n{k}^s\sum _{{d|(k,n)}_r}{\mu }^r\left(d\right)=\sum _{d|n}{d}^s{\mu }^r\left(d\right)\sum _{e=1}^{n/d}{e}^s\hfill \\ & =& \sum _{d|n}{d}^s{\mu }^r\left(d\right)\left(\frac{n^{s+1}}{(s+1)d^{s+1}}+O{(n/d)}^s\right)\hfill \\ & =& \frac{n^{s+1}}{s+1}\sum _{d|n}\frac{{\mu }^r\left(d\right)}{d}+O\left(n^s\sum _{d|n}{\mu }^r\left(d\right)\right)\hfill \\ & =& \frac{n^s}{s+1}{\varphi }_r\left(n\right)+O\left(n^s\right).\hfill \end{array}$$

So, we can calculate

$$\begin{array}{cc}\hfill & \sum _{n\le x}\sum _{k=1}^n{\left({[k,n]}_r\right)}^a\hfill \\ \hfill =& \sum _{n\le x}{n}^a\sum _{d^r|n}\left(\frac{1}{a+1}{\left(\frac{n}{d^r}\right)}^a{\varphi }_r\left(\frac{n}{d^r}\right)+O\left({\left(\frac{n}{d^r}\right)}^a\right)\right)\hfill \\ \hfill =& \sum _{n\le x}\frac{n^a}{a+1}\sum _{d^r|n}\frac{n^a}{d^{ra}}{\varphi }_r\left(\frac{n}{d^r}\right)+O\left(\sum _{n\le x}{n}^{2a}\sum _{d^r|n}\frac{1}{d^r}\right)\hfill \\ \hfill =& \frac{1}{a+1}\sum _{n\le x}\sum _{d^r|n}{d}^{ra}{\left(\frac{n}{d^r}\right)}^{2a}{\varphi }_r\left(\frac{n}{d^r}\right)+O\left(\sum _{n\le x}\sum _{d^r|n}{d}^{ra}{\left(\frac{n}{d^r}\right)}^{2a}\right)\hfill \\ \hfill =& \frac{1}{a+1}\sum _{d^r\le x}{d}^{ra}\sum _{e\le x/d^r}{e}^{2a}{\varphi }_r\left(e\right)+O\left(\sum _{d^r\le x}{d}^{ra}\sum _{e\le x/d^r}{e}^{2a}\right)\hfill \\ \hfill =& \frac{1}{a+1}\sum _{d^r\le x}{d}^{ra}·\frac{1}{2(a+1)\zeta \left(2r\right)}{\left(\frac{x}{d^r}\right)}^{2a+2}+\frac{1}{a+1}\sum _{d^r\le x}{d}^{ra}·O\left({\left(\frac{x}{d^r}\right)}^{2a+1}\log \left(\frac{x}{d^r}\right)\right)\hfill \\ \hfill & +\left(\sum _{d^r\le x}{d}^{ra}{\left(\frac{x}{d^r}\right)}^{2a+1}\right)\hfill \\ \hfill =& \frac{x^{2a+2}}{2{(a+1)}^2\zeta \left(2r\right)}\sum _{d^r\le x}\frac{1}{d^{ar+2r}}+O\left(x^{2a+1}\log x\sum _{d^r\le x}\frac{1}{d^{ar+r}}\right)+O\left(x^{2a+1}\sum _{d^r\le x}\frac{1}{d^{ar+r}}\right)\hfill \\ \hfill =& \frac{x^{2a+2}}{2{(a+1)}^2\zeta \left(2r\right)}\sum _{d=1}^{\infty }\frac{1}{d^{ar+2r}}+O\left(x^{2a+2}\sum _{d>x^{1/r}}\frac{1}{d^{ar+2r}}\right)+O\left(x^{2a+1}\log x\right)\hfill \\ \hfill =& \frac{\zeta (ar+2r)x^{2a+2}}{2{(a+1)}^2\zeta \left(2r\right)}+O\left(x^{2a+1}\log x\right),\hfill \end{array}$$

where we have used the following asymptotic formula:

$$\begin{array}{cc}\hfill \sum _{n\le x}{n}^{2a}{\varphi }_r\left(n\right)=& \sum _{n\le x}{n}^{2a}\sum _{d|n}{\mu }^r\left(d\right)\frac{n}{d}\hfill \\ \hfill =& \sum _{d\le x}{\mu }^r\left(d\right)d^{2a}\sum _{e\le x/d}{e}^{2a+1}\hfill \\ \hfill =& \sum _{d\le x}{\mu }^r\left(d\right)d^{2a}\left(\frac{1}{2a+2}{\left(\frac{x}{d}\right)}^{2a+2}+O{\left(\frac{x}{d}\right)}^{2a+1}\right)\hfill \\ \hfill =& \frac{x^{2a+2}}{2(a+1)}\sum _{d=1}^{\infty }\frac{{\mu }^r\left(d\right)}{d^2}+O\left(x^{2a+2}\sum _{d>x}\frac{{\mu }^r\left(d\right)}{d^2}\right)+O\left(x^{2a+1}\sum _{d\le x}\frac{1}{d}\right)\hfill \\ \hfill =& \frac{x^{2a+2}}{2(a+1)\zeta \left(2r\right)}+O\left(x^{2a+1}\log x\right).\hfill \end{array}$$

This completes the proof of Theorem 4. □

3.4. Proof of Theorem 5

Proof of Theorem 5. 

It is obvious that

$$\begin{array}{cc}\hfill \sum _{k=1}^n\frac{1}{{[k,n]}_r}=& \sum _{k=1}^n\frac{{(k,n)}_r}{kn}=\frac{1}{n}\underset{{(k,n)}_r=d^r}{\sum _{k=1}^n}\frac{d^r}{k}\hfill \\ \hfill =& \frac{1}{n}\sum _{d^r|n}{d}^r\underset{{(k,n/d^r)}_r=1}{\sum _{k=1}^{n/d^r}}\frac{1}{k{d}^r}=\frac{1}{n}\sum _{d^r|n}\underset{{(k,n/d^r)}_r=1}{\sum _{k=1}^{n/d^r}}\frac{1}{k},\hfill \end{array}$$

so by using the properties of the function ${\mu }^r$, we have

$$\begin{array}{cc}\hfill \sum _{n\le x}\sum _{k=1}^n\frac{1}{{[k,n]}_r}=& \sum _{d^r\le x}\frac{1}{d^r}\sum _{h\le x/d^r}\frac{1}{h}\underset{{(k,h)}_r=1}{\sum _{k\le h}}\frac{1}{k}\hfill \\ \hfill =& \sum _{d^r\le x}\frac{1}{d^r}\sum _{h\le x/d^r}\frac{1}{h}\sum _{k\le h}\frac{1}{k}\sum _{{a|(k,h)}_r}{\mu }^r\left(a\right)\hfill \\ \hfill =& \sum _{d^r\le x}\frac{1}{d^r}\sum _{h\le x/d^r}\frac{1}{h}\sum _{a|h}\frac{{\mu }^r\left(a\right)}{a}\sum _{b\le h/a}\frac{1}{b}\hfill \\ \hfill =& \sum _{d^r\le x}\frac{1}{d^r}\sum _{a\le x/d^r}\frac{{\mu }^r\left(a\right)}{a^2}\sum _{c\le x/d^{r}a}\frac{1}{c}\sum _{b\le c}\frac{1}{b}\hfill \\ \hfill =& \sum _{d^r\le x}\sum _{a{d}^r\le x}\frac{1}{d^r}\frac{{\mu }^r\left(a\right)}{a^2}\sum _{c\le x/d^{r}a}\frac{1}{c}\sum _{b\le c}\frac{1}{b}\hfill \\ \hfill =& \sum _{n\le x}\frac{1}{n^2}\sum _{d^r|n}{d}^r{\mu }^r\left(\frac{n}{d^r}\right)\sum _{c\le x/n}\frac{1}{c}\sum _{b\le c}\frac{1}{b}\hfill \\ \hfill =& \sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}\sum _{c\le x/n}\frac{1}{c}\sum _{b\le c}\frac{1}{b}\hfill \\ \hfill =& \sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}\sum _{c\le x/n}\frac{1}{c}\left(\log c+\gamma +O\left(\frac{1}{c}\right)\right)\hfill \\ \hfill =& \sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}\left[\frac{1}{2}{\left(\log \frac{x}{n}\right)}^2+\gamma \log \frac{x}{n}+O\left(1\right)\right]\hfill \\ \hfill =& \frac{1}{2}\sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}{\left(\log \frac{x}{n}\right)}^2+\gamma \sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}\log \frac{x}{n}+O\left(\sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}\right).\hfill \end{array}$$

(20)

Moreover, similar to the proof of (19), we can obtain

$$\begin{array}{cc}\hfill \sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}=& \sum _{d\le x}\frac{{\mu }_r\left(d\right)}{d^2}\sum _{e\le x/d}\frac{1}{e}=\sum _{d\le x}\frac{{\mu }_r\left(d\right)}{d^2}\left(\log \frac{x}{d}+\gamma +O\left(\frac{d}{x}\right)\right)\hfill \\ \hfill =& \left(\log x+\gamma \right)\sum _{d\le x}\frac{{\mu }_r\left(d\right)}{d^2}-\sum _{d\le x}\frac{{\mu }_r\left(d\right)\log d}{d^2}+O\left(\frac{1}{x}\sum _{d\le x}\frac{1}{d}\right)\hfill \\ \hfill =& \left(\log x+\gamma \right)\sum _{d=1}^{\infty }\frac{{\mu }_r\left(d\right)}{d^2}+O\left(\sum _{d>x}\frac{{\mu }_r\left(d\right)}{d^2}\right)-\sum _{d=1}^{\infty }\frac{{\mu }_r\left(d\right)\log d}{d^2}\hfill \\ \hfill & +O\left(\sum _{d>x}\frac{{\mu }_r\left(d\right)\log d}{d^2}\right)+O\left(\frac{1}{x}\sum _{d\le x}\frac{1}{d}\right)\hfill \\ \hfill =& \frac{\log x+\gamma }{\zeta \left(2r\right)}-\frac{2{\zeta }^{{}^{\prime }}\left(2r\right)}{{\zeta }^2\left(2r\right)}+O\left(\frac{\log x}{x}\right).\hfill \end{array}$$

(21)

It follows from (21) and Abel’s identity that

$$\begin{array}{cc}\hfill \sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}\log \frac{x}{n}=& \log x\sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}-\sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}\log n\hfill \\ \hfill =& \frac{1}{\zeta \left(2r\right)}\int \log td\log t+\frac{\gamma }{\zeta \left(2r\right)}\int d\log t\hfill \\ \hfill & -\frac{2{\zeta }^{{}^{\prime }}\left(2r\right)}{{\zeta }^2\left(2r\right)}\int d\log t+O\left(\int \frac{\log t}{t}d\log t\right)\hfill \\ \hfill =& \frac{{\log }^{2}x}{2\zeta \left(2r\right)}+\frac{\gamma \log x}{\zeta \left(2r\right)}-\frac{2{\zeta }^{{}^{\prime }}\left(2r\right)\log x}{{\zeta }^2\left(2r\right)}+O\left(1\right),\hfill \end{array}$$

(22)

and

$$\begin{array}{c}\hfill \sum _{n\le x}\frac{{\varphi }_r\left(n\right)}{n^2}{\left(\log \frac{x}{n}\right)}^2=\frac{{\log }^{3}x}{3\zeta \left(2r\right)}+\frac{\gamma {\log }^{2}x}{\zeta \left(2r\right)}-\frac{2{\zeta }^{{}^{\prime }}\left(2r\right){\log }^{2}x}{{\zeta }^2\left(2r\right)}+O\left(\log x\right).\end{array}$$

(23)

Hence, applying (21)–(23) to (20) gives that

$$\sum _{n\le x}\sum _{k=1}^n\frac{1}{{[k,n]}_r}=\frac{{\left(\log x\right)}^3}{6\zeta \left(2r\right)}+A{\left(\log x\right)}^2+O\left(\log x\right),$$

where $A=\frac{\gamma }{\zeta \left(2r\right)}-\frac{{\zeta }^{{}^{\prime }}\left(2r\right)}{{\zeta }^2\left(2r\right)}$.

This completes the proof of Theorem 5. □

4. Conclusions

In this paper, we firstly constructed various asymptotic formulas for the average of r-gcd-sum function and r-lcm-sum function in Theorems 1, 3 and 4, and the sums of reciprocals of r-gcd and r-lcm were also studied in Theorems 2 and 5. Additionally, Corollary 2 gives results on the order of magnitude of certain sums concerning the r-gcd’s and r-lcm’s of two positive integers. These results present the different distributional properties for the generalizations of usual gcd function and lcm function. Furthermore, we constructed two asymptotic formulas in Theorems 3 and 4 with a positive integer $a\ge 2$, and the study did not include the results with $a\le -2$, so that will be the focus of our upcoming research. More generally, let $P_{f,r}\left(n\right)={\sum }_{k=1}^{n}f\left(gcd{(k,n)}_r\right)$ with an arbitrary arithmetical function f. The weighted averages of the r-gcd-sum functions with any mathematical function f is one of our future research directions, where the weights are divisor functions, Bernoulli polynomials, gamma functions, etc.