Menon's identity states that for every positive integer n one has∑(a− 1, n)= φ (n) τ (n), where
a runs through a reduced residue system (mod n),(a− 1, n) stands for the greatest common …

1. Motivation and main result Page 1 TAIWANESE JOURNAL OF MATHEMATICS Vol. 23, No.
3, pp. 557–561, June 2019 DOI: 10.11650/tjm/180904 Short Proof and Generalization of a …

The classical Menon's identity [PK Menon, On the sum∑(a− 1, n)[(a, n)= 1], J. Indian Math.
Soc.(NS) 29 (1965) 155–163] states that∑ a∈ ℤ n∗ gcd (a− 1, n)= φ (n) σ 0 (n), where for a …

Let n be a positive integer. Arai-Carlitz's identity is ∑ _ a, b, a+ b ∈ (Z/n Z)^ *\gcd (a+ b-1,
n)= X (n) τ (n),∑ a, b, a+ b∈(Z/n Z)× gcd (a+ b-1, n)= X (n) τ (n), where X (n)= ∑ _ a, b, a+ b …

The classical Menon's identity [7] states that∑ a= 1 gcd⁡(a, n)= 1 n gcd⁡(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 …

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∑ …

The $ k $-dimensional generalized Euler function $\varphi_k (n) $ is defined to be the
number of ordered $ k $-tuples $(a_1, a_2,\ldots, a_k)\in\mathbb {N}^ k $ with $1\leq a_1 …

Let A= Fq [T] be the polynomial ring over the finite field Fq. In this paper, we prove a Menon–
Sury's identity with several multiplicative and additive characters for any arithmetic function f …

We introduce and study the generalized cyclotomic polynomials ΦA, S, n (x) associated with
a regular system A of divisors and an arbitrary set S of positive integers. We show that all of …

Abstract Let $\varphi (n) $ be Euler's totient function and $\tau (n) $ the divisor function.
Recently, Zhao and Cao proved that $$\sum _ {\substack {a= 1\atop\gcd (a, n)= 1}}^{n}\gcd …