## Taiwanese Journal of Mathematics

### A Menon-type Identity with Multiplicative and Additive Characters

#### Abstract

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

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

Dates
Revised: 14 June 2018
Accepted: 10 July 2018
First available in Project Euclid: 12 July 2018

https://projecteuclid.org/euclid.twjm/1531382426

Digital Object Identifier
doi:10.11650/tjm/180702

Mathematical Reviews number (MathSciNet)
MR3952238

Zentralblatt MATH identifier
07068561

#### Citation

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

#### References

• P. Haukkanen, Menon's identity with respect to a generalized divisibility relation, Aequationes Math. 70 (2005), no. 3, 240–246.
• P. Haukkanen and J. Wang, A generalization of Menon's identity with respect to a set of polynomials, Portugal. Math. 53 (1996), no. 3, 331–337.
• ––––, High degree analogs of Menon's identity, Indian J. Math. 39 (1997), no. 1, 37–42.
• Y. Li, X. Hu and D. Kim, A generalization of Menon's identity with Dirichlet characters, to appear in Int. J. Number Theory.
• Y. Li and D. Kim, A Menon-type identity with many tuples of group of units in residually finite Dedekind domains, J. Number Theory 175 (2017), 42–50.
• ––––, Menon-type identities derived from actions of subgroups of general linear groups, J. Number Theory 179 (2017), 97–112.
• P. K. Menon, On the sum $\sum (a-1,n) [(a,n)=1]$, J. Indian Math. Soc. (N.S.) 29 (1965), 155–163.
• C. Miguel, Menon's identity in residually finite Dedekind domains, J. Number Theory 137 (2014), 179–185.
• ––––, A Menon-type identity in residually finite Dedekind domains, J. Number Theory 164 (2016), 43–51.
• I. M. Richards, A remark on the number of cyclic subgroups of a finite group, Amer. Math. Monthly 91 (1984), no. 4, 571–572.
• V. Sita Ramaiah, Arithmetical sums in regular convolutions, J. Reine Angew. Math. 303/304 (1978), 265–283.
• R. Sivaramakrishnan, A number-theoretic identity, Publ. Math. Debrecen 21 (1974), 67–69.
• B. Sury, Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo (2) 58 (2009), no. 1, 99–108.
• M. Tărnăuceanu, A generalization of Menon's identity, J. Number Theory 132 (2012), no. 11, 2568–2573.
• L. Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Semin. Mat. Univ. Politec. Torino 69 (2011), no. 1, 97–110.
• ––––, Menon-type identities concerning Dirichlet characters, Int. J. Number Theory 14 (2018), no. 4, 1047–1054.
• X.-P. Zhao and Z.-F. Cao, Another generalization of Menon's identity, Int. J. Number Theory 13 (2017), no. 9, 2373–2379.