1. Introduction
For any integer
, the Pillai’s arithmetical function, which is also known as the gcd-sum function is defined by
where
is the Euler’s totient function, and
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
was proved by Bordellés [
1], namely, for every
,
where
is Euler’s constant and
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
On the other hand, let
denote the least common multiple of
k and
n. The lcm-sum function is defined as
for any integer
. Bordellés investigated the function
in his article [
2] and established that
where
is the Glaisher–Kinkelin constant.
Let
be a fixed integer, the greatest
power common divisor of positive integers
a and
b is defined to be the largest positive integer
such that
and
, which is denoted by
and called the
r-gcd of
a and
b. Note that
. The related research on greatest
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
is visible along the the curve
if and only if
, where
and
. So the immediate natural extension is to consider visibility along curves
or
. It is obvious that the
r-gcd function provides a necessary criterion for the combined visibility of a lattice point along the curves
or
. 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:
which is called r-gcd-sum function. They obtained an asymptotic formula for its summatory function:
With the help of (
8), the authors [
5] studied the partial sums of the Dirichlet series of the r-gcd-sum function, that is
for all real values of
t and any integer
, and established asymptotic formulas for this function. Moreover, they [
6] also introduced an
r-gcd-sum function over
r-regular integers
, which is defined by
where
= {
,
k is
r-regular mod
}. Based on the properties of a generalization of Euler’s
-function [
3], they obtained some arithmetic properties of
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
power common multiple as
which is also called
r-lcm of
k and
n. They proved that
with positive integer
.
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 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 -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 and integer , we have Notice that for
, the asymptotic formula for average of the usual gcd-sum function
had been done in [
1] by the convolution identity
, where
and ∗ is the usual Dirichlet convolution product.
Theorem 2. For any real number , we have the following estimate When
, the main term of estimate (
11) is equal to the main term of (
3). Moreover, the error term of estimate (
3) is improved from
to
because of the classical result of Walfisz [
8],
Theorem 3. Let be a positive integer, then we obtain Theorem 4. Let a be a positive integer, then for any real number sufficiently large, we have the following estimate Here we offer a more simple proof of by using elementary calculations.
Taking in Theorem 4, we can get
Corollary 1. For any real number sufficiently large, Theorem 5. For any real number , we havewhere.
We know that for any function
, there is the general identity
which is symmetric in the variables. So it is easy to get the following asymptotic formulas from (
10), (
11), (
14) and (
15).
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 , and the study did not include the results with , so that will be the focus of our upcoming research. More generally, let 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.