Open Access
June, 2019 A Menon-type Identity with Multiplicative and Additive Characters
Yan Li, Xiaoyu Hu, Daeyeoul Kim
Taiwanese J. Math. 23(3): 545-555 (June, 2019). DOI: 10.11650/tjm/180702


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].


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Yan Li. Xiaoyu Hu. Daeyeoul Kim. "A Menon-type Identity with Multiplicative and Additive Characters." Taiwanese J. Math. 23 (3) 545 - 555, June, 2019.


Received: 13 February 2018; Revised: 14 June 2018; Accepted: 10 July 2018; Published: June, 2019
First available in Project Euclid: 12 July 2018

zbMATH: 07068561
MathSciNet: MR3952238
Digital Object Identifier: 10.11650/tjm/180702

Primary: 11A07 , 11A25

Keywords: additive character , Chinese remainder theorem , Dirichlet character , divisor function , Euler's totient function , Iverson bracket , Menon's identity

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

Vol.23 • No. 3 • June, 2019
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