Asymptotic formulas for generalized gcd-sum and lcm-sum functions over $r$-regular integers $(\bmod\ n^{r})$

1.   Introduction and main results

For any integer $n\geq1$, the Pillai's arithmetical function, which is also known as the gcd-sum function is defined by

where $\varphi$ is the Euler's totient function, and $(k, n)$ denotes the greatest common divisor of $k$ and $n$. Over the years, a great deal of mathematical effort in number theory has been devoted to the study of classical gcd-sum function. Its distribution properties, numerous important arithmetical and algebraic information have been investigated by many mathematicians; see, for example ^[1,2,3]. In addition, some natural generalizations of the usual gcd-sum function also have been considered; see ^[4]. Let $r\geq1$ be a fixed integer, the greatest $r^{th}$ power common divisor of positive integers $a$ and $b$ is defined to be the largest positive integers $d^{r}$ 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 $(a, b)_{1} = (a, b)$. With this definition, Prasad, Reddy and Rao defined the $r$-gcd-sum function as

and for every $\varepsilon > 0$, they derived an asymptotic formula of its summatory function as follow:

where $\zeta(n)$ is the Riemann zeta function and $\theta$ is the positive real number appearing in the Dirichlet divisor problem, for which one seeks the smallest positive real number $\theta$. That is, for all real $x\geq1$ and any $\varepsilon > 0$, the asymptotic formula given below holds:

where $\tau(n)$ is the number of positive divisors of $n$ and $\gamma$ is Euler's constant.

Let $r$ be a fixed positive integer. A positive integer $k$ is said to be $r$-regular $(\bmod\ n^{r})$ if there exists an integer $x$ such that $k^{r+1}x\equiv k^{r}$ $(\bmod\ n^{r})$. Also, it was observed in ^[5] that $k$ is $r$-regular $(\bmod\ n^{r})$ if and only if $\left(k, n^{r}\right)_{r}$ is a unitary divisor of $n^{r}$. We recall that $d$ is said to be a unitary divisor of $n$ if $d|n$ and $(d, n/d) = 1$, written as $d\parallel n$. When $r = 1$, it gives usual regular integers $(\bmod\ n)$, and a detailed study was initiated by L. Tóth ^[6]. Let $Reg_{r}(n)$ denotes the set of all $r$-regular integers modulo $n$ and $Reg(n)$ denote the set of all regular integers modulo $n$. That is, $Reg_{r}(n^{r})$ = $\{ k:1\leq k\leq n^{r}$, $k$ is $r$-regular mod $n^{r} \}$, $Reg(n)$ = $\{ k:1\leq k\leq n$, $k$ is regular mod $n \}$. L. Tóth ^[7] introduced another generalized gcd-sum function over regular integers modulo $n$ as

He also proved that $\tilde{P}(n)$ is multiplicative and gave the following asymptotic formula

where $\delta(x) = \exp\left(-C(\log x)^{3/5}(\log \log x)^{-1/5}\right)$, and $K_{1}$, $K_{2}$ are given by

where $\psi(n)$ is Dedekind function defined by $\psi(n) = n\prod_{p\mid n}\left(1+\frac{1}{p}\right)$ and

Recently, under the Riemann hypothesis, the error term $R(x)$ in (1.4) has been improved by Zhang and Zhai ^[8] to

where $\varepsilon > 0$ is any sufficiently small positive number.

Afterwards, Prasad, Reddy and Rao ^[9] introduced the $r$-gcd-sum function over $r$-regular integers $(\bmod\ n^{r})$, which is defined as

Based on the properties of a generalization of Euler's $\varphi$-function ^[10], they obtain some arithmetic properties of $\tilde{P_{r}}(n^{r})$ and an asymptotic formula for its summatory function. Motivated and inspired by the work of L.Tóth, Prasad, Reddy and Rao ^[7,9], in this paper we offer a different proof for Prasad, Reddy and Rao's result ^[9] firstly, then we perform a further investigation for $r$-gcd-sum function over $r$-regular integers $(\bmod\ n^{r})$ and derive two kinds of asymptotic formulas. Furthermore, we also establish estimates for the generalized $r$-lcm-sum function over $r$-regular integers $(\bmod\ n)$.

Firstly, we state the following results.

Theorem 1. For any real number $x > e$ sufficiently large, we have the following estimate

where

Note. Taking $r = 1$, L. Tóth's result (^[7], Theorem 2) can be deduced from Theorem 1.

Theorem 2. For a positive integer $s > 1$, define the generalized $r$-gcd-sum function over $r$-regular integers $(\bmod\ n^{r})$ as

Then for any real number $x > e$ sufficiently large and $r > 1$, we have

On the other hand, let $[k, n]$ denotes the least common multiple of $k$ and $n$. Ikeda and Matsuoka ^[11] studied the functions

and obtained

for $a\in \mathbb{N}$. So naturally, we're thinking about generalized $r$-lcm-sum function here. Define the least $r^{th}$ power common multiple as

and for a positive integer $a > 1$, and generalized $r$-lcm-sum function over $r$-regular integers $(\bmod\ n)$ as

Then we can deduce,

Theorem 3. For any real number $x > e$ sufficiently large, define the $r$-lcm-sum function as $L_{r, a}(n) = \sum_{k = 1}^{n}\left([k, n]_{r}\right)^{a}$, then

Theorem 4. For any real number $x > e$ sufficiently large, we obtain

Taking $r = 1$ in Theorem 3, we can get,

Corollary 1. For any real number $x > e$ sufficiently large,

2.   Several lemmas

In this section, we give several lemmas which are necessary in the proof of our theorems. First of all, in the proofs of these lemmas, we need some knowledge of elementary and analytic number theory. And we know that an integer is $r$th-power-free if it is not divisible by the $rth$ power of any integer $> 1$. Then let $\phi_{r}(n)$ denotes the number of integers $k$ in the set $1, \cdots, n$, for which the greatest common divisor $(k, n)$ is $r$th-power-free, and $\mu^{r}(n)$ is defined as follows:

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

and let $\tau^{*}(n; m)$ denotes the number of unitary divisors of $n$ which are relatively prime to $m$, where $m\geq 1$ be an integer. In symbols,

By the notations above, we have the following:

Lemma 1. For every $\varepsilon > 0$, we get

where $\sigma^{*}_{a}(n)$ is the sum of the a-th power of square free divisors of $n$ and $\alpha(m)$, $\beta(m)$, $\psi(m)$, $\delta(m)$ are the same as given in the introduction.

Proof. For any integer $m\geq 1$ and every $\varepsilon > 0$, one knows that (^[12], Theorem 4.3),

Then by using partial summation formula, we immediately have

Lemma 2. (^[13], Corollary 2.1) For any integer $k\geq1$ and $s\geq0$

where $\tau(k)$ denotes the number of divisors of $k$.

Lemma 3. For every $n, a, r\in\mathbb{N}$, we have the identity

where $g_{0}(n) = 1$, if $n$ is $r$th-power-free; 0, otherwise. And $B_{n}$ are the Bernoulli numbers defined by exponential generating function $\frac{t}{e^{t}-1} = \sum_{n = 0}^{\infty}B_{n}\frac{t^{n}}{n!}$.

Proof. Firstly, we know the function $\mu^{r}$ has following property ^[10]:

In addition, it is well known that for every $n, a\in\mathbb{N}$,

So we can deduce

This completes the proof of Lemma 3.

Lemma 4. For any integer $k\geq1$ and $a\in\mathbb{N}$, we have

where $\varphi_{r}$ is Jordan totient function, defined by

Proof. It follows from Lemma 2 that

where we use the familiar estimate $\sum_{n\leq x}\frac{1}{n} = O(\log x)$ and $\sum_{n > x}\frac{1}{n^{s}} = O\left(x^{1-s}\right)$, when $s > 1$. Furthermore, it can be deduced

So we have

By using similar methods and elementary asymptotic formula

(b) can be easily derived.

3.   The proof of Theorem 1

Let $R_{n, r}$ = $\{ k:1\leq k\leq n^{r}$, $\left(k, n^{r}\right)_{r} = 1\}$ and $\phi_{r}\left(n^{r}\right)$ denote the number of elements in $R_{n, r}$. Since $\phi_{r}\left(n^{r}\right) = n^{r}\sum_{d\mid n}\mu(d)/d^{r}$ (see, ^[14]), we have

Taking $y = x^{1/r}$, it follows from Lemma 1 and the above identity that

Notice that for any integer $n\geq1$,

which implies that $\alpha(n) = O(\log(n))$, $\beta(n) = O(1)$.

Then by the definition of $\psi$, it is clear that for any integer $r\geq0$, the series

and

are both absolutely convergent. Applying Euler product, we can obtain

Moreover, since $\sigma^{*}_{-\theta}(n)\leq\tau(n)$, $\sigma^{*}_{-1+\varepsilon}(n)\leq\tau(n)$ for $\varepsilon\leq1$, and $y^{r}\delta (y)$ is increasing. so the error term can be simplified as

Now, by a simple calculation, we get

This completes the proof of Theorem 1.

4.   The proof of Theorem 2

For any integer $r > 1$, $s > 1$, by taking $y = x^{1/r}$ and using lemma 2, we have

Since $r > 1$, $s > 1$, the series

is absolutely convergent and the general term is a multiplicative function, thus the series can be expended into an infinite product of Euler type

For the error term, we can deduce that

To sum up, we can obtain

5.   The proof of Theorems 3 and 4

Firstly, we prove Theorem 4. By using Lemma 3, it is easily seen that

Then

By using Lemma 4(a), we get

And it can be calculated by Euler product that

So

where we have used

And for $m, a\in{\rm I\!N}$ with $a > m$,

so

To sum up, we can obtain that

This completes the proof of Theorem 4.

Theorem 3 can be deduced by Lemma 3 and Lemma 4(b), and the proof method is similar to Theorem 4.

6.   Conclusions

In this article, we firstly introduced the basic concepts and properties of $r$-regular integers $(\bmod\ n^{r})$, and the $r$-gcd-sum function and $r$-lcm-sum function over those integers. Let $\tilde{P_{r}}(n^{r})$ denote the $r$-gcd-sum function over $r$-regular integers $(\bmod\ n^{r})$, Theorem 1 offered a different proof of Prasad, Reddy and Rao's result, and L. Tóth's result (^[7], Theorem 2) can be deduced from our result. Theorem 2 gave an asymptotic formula for generalized r-gcd-sum function over $r$-regular integers $(\bmod\ n^{r})$. In addition, we defined least $r$th power common multiple $[k, n]_{r}$ based on the relationship between the common lcm and gcd, we also defined $r$-lcm-sum function $L_{r, a}(n)$ and generalized $r$-lcm-sum function $\tilde{L}_{r, a}(n)$ over $r$-regular integers $(\bmod\ n)$. By making use of Euler product and some knowledge of elementary and analytic number theory, we obtained asymptotic formulas for their summatory function. As a supplement, we can also got an asymptotic formula for lcm-sum function over regular integers $(\bmod\ n)$ easily.

Acknowledgement

The authors would like to thank the referees for their very helpful and detailed comments. This work is supported by the N.S.F.(11971381, 11701447, 11871317) and P.N.S.F.(2015KJXX-27) of China.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.