Menon-type Identity Research Articles

A Menon-type identity derived using Cohen-Ramanujan sum

A Menon-type identity derived using Cohen-Ramanujan sum

A regular Tóth identity and a Menon-type identity in residually finite Dedekind domains

A regular Tóth identity and a Menon-type identity in residually finite Dedekind domains

Menon-Type Identities Concerning Subsets of the Set {1, 2,..., n}

We prove certain Menon-type identities associated with the subsets of the set {1, 2,..., n} and related to the functions f, fk, Ф and Ф k, defined and investigated by Nathanson.

A Menon-type identity with Dirichlet characters in residually finite Dedekind domains

A Menon-type identity with Dirichlet characters in residually finite Dedekind domains

A Menon-type identity concerning Dirichlet characters and a generalization of the gcd function

Menon’s identity is a classical identity involving gcd sums and the Euler totient function \(\phi \). In a recent paper, Zhao and Cao (Int. J. Number Theory 13(9) (2017) 2373–2379) derived the Menon-type identity \(\sum _{\begin{array}{c} k=1 \end{array}}^{n}(k-1,n)\chi (k) = \phi (n)\tau (\frac{n}{d})\), where \(\chi \) is a Dirichlet character mod n with conductor d. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao–Cao identity can be improved if one uses the method we employ here.

Menon-type identities with respect to sets of units

Menon-type identities with respect to sets of units

English

Menon’s identity is a classical identity involving gcd sums and the Euler totient function φ. A natural generalization of φ is the Klee’s function Φs. We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).

Another generalization of Euler\u2019s arithmetic function and Menon\u2019s identity

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 1le a_1,ldots ,a_kle n and both the product a_1cdots 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.

Another regular Menon-type identity in residually finite Dedekind domains

We give a regular extension of the Menon-type identity to residually finite Dedekind domains.

Menon-type identities again: A note on a paper by Li, Kim and Qiao

We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple character free approach for the proof.

A Menon-type Identity with Multiplicative and Additive Characters

This paper studies Menon-type identities involving both multiplicative characters and additive characters. In the paper, we shall give the explicit formula of the following sum \[ \sum_{\substack{a \in \mathbb{Z}_n^{\ast} \\ b_1, \ldots, b_k \in \mathbb{Z}_n}} \gcd(a-1, b_1, \ldots, b_k, n) \chi(a) \lambda_1(b_1) \cdots \lambda_k(b_k), \] where for a positive integer $n$, $\mathbb{Z}_n^{\ast}$ is the group of units of the ring $\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z}$, $\gcd$ represents the greatest common divisor, $\chi$ is a Dirichlet character modulo $n$, and for a nonnegative integer $k$, $\lambda_1, \ldots, \lambda_k$ are additive characters of $\mathbb{Z}_n$. Our formula further extends the previous results by Sury [13], Zhao-Cao [17] and Li-Hu-Kim [4].

Short Proof and Generalization of a Menon-type Identity by Li, Hu and Kim

We present a simple proof and a generalization of a Menon-type identity by Li, Hu and Kim, involving Dirichlet characters and additive characters.

Menon-type identities concerning additive characters

By considering even functions (hbox {mod} n), we generalize a recent Menon-type identity by Li and Kim, involving additive characters of the group {mathbb {Z}}_n. We use a different approach, based on certain convolutional identities. Some other applications, including related formulas for Ramanujan sums, are discussed as well.

A regular Menon-type identity in residually finite Dedekind domains

A regular Menon-type identity in residually finite Dedekind domains

Menon-type identities with additive characters

The classical Menon's identity [7] states that∑a=1gcd⁡(a,n)=1ngcd⁡(a−1,n)=φ(n)τ(n),for every positive integer n, where φ(n) is the Euler's totient function and τ(n) is the number of positive divisors of n. Recently, Zhao and Cao [19] extended Menon's identity to Dirichlet characters:∑a=1gcd⁡(a,n)=1ngcd⁡(a−1,n)χ(a)=φ(n)τ(nd),where χ is a Dirichlet character modulo n and d is the conductor of χ. In this paper, we extend Menon's identity to additive characters. A special case of our main result reads like this:∑a=1gcd⁡(a,n)=1ngcd⁡(a−1,n)exp⁡(ka2πin)=exp⁡(k2πin)τ(gcd⁡(k,n))φ(n) if ordp(n)−ordp(k)≠1 holds for any prime p dividing n, where for u∈Z, ordp(u) is the exponent of the highest power of p dividing u. For k=0, our result reduces to the classical Menon's identity.

Menon-type identities concerning Dirichlet characters

Let [Formula: see text] be a Dirichlet character (mod [Formula: see text]) with conductor [Formula: see text]. In a quite recent paper Zhao and Cao deduced the identity [Formula: see text], which reduces to Menon’s identity if [Formula: see text] is the principal character (mod [Formula: see text]). We generalize the above identity by considering even functions (mod [Formula: see text]), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums.

Menon-type identities derived from actions of subgroups of general linear groups

In this article, we consider the actions of subgroups of the general linear group GLr(Zn) on Znr, including groups of upper triangular matrices in GLr(Zn), unipotent groups, Heisenberg groups and extended Heisenberg groups. By applying the Cauchy–Frobenius–Burnside lemma, we obtain several generalizations of the well known Menon's identity.

A Menon-type identity with many tuples of group of units in residually finite Dedekind domains

B. Sury proved the following Menon-type identity,∑a∈U(Zn),b1,⋯,br∈Zngcd⁡(a−1,b1,⋯,br,n)=φ(n)σr(n), where U(Zn) is the group of units of the ring for residual classes modulo n, φ is the Euler's totient function and σr(n) is the sum of r-th powers of positive divisors of n with r being a non-negative integer. Recently, C. Miguel extended this identity from Z to any residually finite Dedekind domain. In this note, we will give a further extension of Miguel's result to the case with many tuples of group of units. For the case of Z, our result reads as follows∑a1,⋯,as∈U(Zn),b1,⋯,br∈Zngcd⁡(a1−1,⋯,as−1,b1,⋯,br,n)=φ(n)∏i=1m(φ(piki)s−1pikir−piki(s+r−1)+σs+r−1(piki)), where n=p1k1⋯pmkm is the prime factorization of n.

A Menon-type identity in residually finite Dedekind domains

Let φ be the Euler's totient function and σk(n)=∑d|ndk, that is, the sum of the kth powers of the divisors of n. B. Sury showed that∑t1∈U(Zn),t2,…,tr∈Zngcd(t1−1,t2,…,tr,n)=φ(n)σr−1(n), where U(Zn) is the group of units in the ring of residual classes modulo n. Here, this identity is extended to residually finite Dedekind domains.