Taiwanese Journal of Mathematics

A Menon-type Identity with Multiplicative and Additive Characters

Yan Li, Xiaoyu Hu, and Daeyeoul Kim

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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].

Article information

Taiwanese J. Math., Volume 23, Number 3 (2019), 545-555.

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

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Zentralblatt MATH identifier

Primary: 11A07: Congruences; primitive roots; residue systems 11A25: Arithmetic functions; related numbers; inversion formulas

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


Li, Yan; Hu, Xiaoyu; Kim, Daeyeoul. A Menon-type Identity with Multiplicative and Additive Characters. Taiwanese J. Math. 23 (2019), no. 3, 545--555. doi:10.11650/tjm/180702. https://projecteuclid.org/euclid.twjm/1531382426

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