Abstract
We define the k-dimensional generalized Euler function \(\varphi _k(n)\) as the number of ordered k-tuples \((a_1,\ldots ,a_k)\in {\mathbb {N}}^k\) such that \(1\le a_1,\ldots ,a_k\le n\) and both the product \(a_1\cdots a_k\) and the sum \(a_1+\cdots +a_k\) are prime to n. We investigate some of the properties of the function \(\varphi _k(n)\), and obtain a corresponding Menon-type identity.
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1 Motivation
Jordan’s arithmetic function \(J_k(n)\) is defined as the number of ordered k-tuples \((a_1,\ldots ,a_k)\in {\mathbb {N}}^k\) such that \(1\le a_1,\ldots ,a_k\le n\) and the gcd \((a_1,\ldots ,a_k,n)=1\). It is well known that \(J_k(n)\) is multiplicative in n and \(J_k(n)=n^k \prod _{p\mid n} (1-1/p^k)\). If \(k=1\), then \(J_1(n)=\varphi (n)\) is Euler’s arithmetic function.
A Menon-type identity concerning the function \(J_k(n)\), obtained by Nageswara Rao [7], is given by
where \(\tau (n)\) is the number of divisors of n. If \(k=1\), then (1.1) reduces to Menon’s original identity [6].
Euler’s arithmetic function and Menon’s identity have been generalized in various directions by several authors. See, e.g., the books [5, 8], the papers [4, 7, 9, 11, 12], and their references.
The function \(X(n)= \# \{(a,b)\in {\mathbb {N}}^2: 1\le a,b\le n, (ab,n)=(a+b,n)=1 \}\) is an analog of Euler’s \(\varphi \)-function, and was introduced by Arai and Gakuen [2]. It was shown by Carlitz [3] that the function X(n) is multiplicative and
Note that if n is even, then \(X(n)=0\). The function X(n) can also be given as
\(\mu \) denoting the Möbius function. The corresponding Menon-type identity
was deduced by Sita Ramaiah [9, Cor. 10.4]. In fact, (1.2) is a corollary of a more general identity involving Narkiwicz-type regular systems of divisors and k-reduced residue systems.
Recently, identity (1.2) was generalized by Ji and Wang [4, 12] to residually finite Dedekind domains, by using Narkiwicz-type regular systems of divisors, and to the ring of algebraic integers, concerning Dirichlet characters modulo n, respectively. Note that in paper [12] identity (1.2) is called the “Arai–Carlitz identity.” However, Arai and Carlitz only considered the function X(n) and did not deduce such an identity. We refer to (1.2) as the Sita Ramaiah identity.
It is natural to introduce and to study the following k-dimensional generalization of the function X(n), and to ask if the corresponding generalization of the Sita Ramaiah identity is true for it. These were not investigated in the literature, as far as we know. For \(k\in {\mathbb {N}}\) we define the function \(\varphi _k(n)\) as
Note that \(\varphi _1(n)=\varphi (n)\) is Euler’s function and \(\varphi _2(n)=X(n)\) of above. We investigate some of the properties of the function \(\varphi _k(n)\), and obtain a corresponding Menon-type identity. Our main results are included in Sect. 2, and their proofs are presented in Sects. 3 and 4.
We will use the following notations: \({\text {id}}_k(n)=n^k\), \(\mathbf{1}(n)=1\) (\(n\in {\mathbb {N}}\)), \(\omega (n)\) will denote the number of distinct prime factors of n, and “\(*\)” the Dirichlet convolution of arithmetic functions.
2 Main results
In this paper we prove the following results.
Theorem 2.1
For every \(k,n\in {\mathbb {N}}\),
It is a consequence of Theorem 2.1 that the function \(\varphi _k(n)\) is multiplicative. Also, \(\varphi _k(n)=0\) if and only if k and n are both even. Further properties of \(\varphi _k(n)\) can be deduced. Its average order is given by the next result.
Theorem 2.2
Let \(k\ge 2\) be fixed. Then
where
Corollary 2.3
(\(k=2\)) We have
where
We have the following generalization of Menon’s identity.
Theorem 2.4
Let f be an arbitrary arithmetic function. Then for every \(k,n\in {\mathbb {N}}\),
Corollary 2.5
(\(f(n)=n\)) For every \(k,n\in {\mathbb {N}}\),
If \(k=1\), then (2.4) reduces to Menon’s identity and if \(k=2\), then it gives the Sita Ramaiah identity (1.2).
3 Proofs of Theorems 2.1 and 2.2
We need the following lemmas.
Lemma 3.1
Let \(n,d\in {\mathbb {N}}\), \(d\mid n\), and let \(r\in {\mathbb {Z}}\). Then
Lemma 3.1 is known in the literature, usually proved by the inclusion–exclusion principle. See, e.g., [1, Th. 5.32]. The following generalization and a different approach of proof are given in our paper [11].
Lemma 3.2
[11, Lemma 2.1] Let \(n,d,e\in {\mathbb {N}}\), \(d\mid n\), \(e\mid n\) and let \(r,s\in {\mathbb {Z}}\). Then
In the case \(e=1\), Lemma 3.2 reduces to Lemma 3.1.
We need to define the following slightly more general function than \(\varphi _k(n)\):
If \(m=n\), then \(\varphi _k(n,n)=\varphi _k(n)\), given by (1.3).
Lemma 3.3
(recursion formula for \(\varphi _k(n,m)\)) Let \(k\ge 2\) and \(m\mid n\). Then
Proof of Lemma 3.3
We have
By using Lemma 3.1 we deduce that
where \(d\mid m\) and \(m\mid n\) imply that \(d\mid n\). \(\square \)
Proof of Theorem 2.1
Let \(k,n,m\in {\mathbb {N}}\) such that \(m\mid n\). We show that
By induction on k. If \(k=1\), then \(\varphi _1(n,m)= \varphi (n)\), by its definition (3.1). Let \(k\ge 2\). Assume that (3.2) holds for \(k-1\) and prove it for k. We have, by using Lemma 3.3,
which proves formula (3.2). Now choosing \(m=n\), (3.2) gives identity (2.1), which can be rewritten as (2.2). \(\square \)
Proof of Theorem 2.2
Let \(\varphi _k= {\text {id}}_k * g_k\), that is, \(g_k=\varphi _k * \mu {\text {id}}_k\). Here the function \(g_k(n)\) is multiplicative and for any prime power \(p^\nu \) (\(\nu \ge 1\)),
We obtain from (2.2) that for \(\nu \ge 2\),
and for \(\nu =1\),
a polynomial in p of degree \(k-1\), with leading coefficient \(-(k+1)\). Actually, we have
for every integer \(k\ge 2\) and every prime \(p\ge 2\). To see this, note that by Lagrange’s mean value theorem,
and from (3.4) we deduce that
On the other hand, \(p^k-(p-1)^k>1\), \((p-1)^k-(-1)^k<p^k\) imply that
According to (3.5), \(|g_k(p)|< (k+1)p^{k-1}\) holds true for every \(k\ge 2\) and every \(p\ge 2\), and by (3.3) we deduce that
To obtain the desired asymptotic formula we apply elementary arguments. We have
Here the main term is \(\frac{C_k}{k+1} x^{k+1}\) by using the Euler product formula. To evaluate the error terms consider the Piltz divisor function \(\tau _{k+1}(n)\), representing the number of ordered \((k+1)\)-tuples \((a_1,\ldots ,a_{k+1})\in {\mathbb {N}}^{k+1}\) such that \(a_1\cdots a_{k+1}=n\). We have \(\tau _{k+1}(p^\nu )\ge \tau _{k+1}(p)=k+1\) for every prime power \(p^{\nu }\) (\(\nu \ge 1\)), and \(\tau _{k+1}(n)\ge (k+1)^{\omega (n)}\) for every \(n\in {\mathbb {N}}\).
We obtain
and
by using known elementary estimates on the Piltz divisor function. See, e.g., [10, Lemma 3]. This completes the proof. \(\square \)
4 Proof of Theorem 2.4
Let \(M_k(n)\) denote the sum on the left-hand side of (2.3). We have by the convolutional identity \(f=(\mu *f)*\mathbf{1}\),
that is
where
Next we evaluate the sum \(N_k(n,d,\delta )\), where \(d\mid n\), \(\delta \mid n\) are fixed. If \((d,\delta )>1\), then \(N_k(n,d,\delta )=0\), the empty sum. So, assume that \((d,\delta )=1\). If \(k=1\), then by using Lemma 3.2 we deduce
since for each term of the sum \(\delta \mid a_1\) and \(\delta \mid n\), which gives \(\delta \mid (a_1,n)=1\), so \(\delta =1\).
Lemma 4.1
(Recursion formula for \(N_k(n,d,\delta )\)) Let \(k\ge 2\), \(d\mid n\), \(\delta \mid n\), \((d,\delta )=1\). Then
Proof of Lemma 4.1
We have
Using that \((d,\delta )=1\) and applying Lemma 3.2 we deduce that
\(\square \)
Lemma 4.2
Let \(k\ge 2\), \(d\mid n\), \(\delta \mid n\), \((d,\delta )=1\). Then
Proof of Lemma 4.2
By induction on k. If \(k=2\), then by the recursion (4.3) and (4.2),
Hence, the formula is true for \(k=2\). Assume it holds for \(k-1\), where \(k\ge 3\). Then we have, by the recursion (4.3),
where the condition \((t,j)=1\) can be omitted, since \(j\mid d\), \(t\mid \delta \) and \((d,\delta )=1\). We deduce that
giving (4.4), which completes the proof of Lemma 4.2. \(\square \)
Now we continue the evaluation of \(M_k(n)\). According to (4.1) and Lemma 4.2, we have
where the inner sum is
This leads to
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Tóth, L. Another generalization of Euler’s arithmetic function and Menon’s identity. Ramanujan J 57, 811–822 (2022). https://doi.org/10.1007/s11139-020-00353-z
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DOI: https://doi.org/10.1007/s11139-020-00353-z