Another regular Menon-type identity in residually finite Dedekind domains
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ANOTHER REGULAR MENON-TYPE IDENTITY IN RESIDUALLY FINITE DEDEKIND DOMAINS CH. JI1,∗ and Y. WANG2 1
2
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P. R. China e-mail: [email protected]
School of Mathematics and Statistics, Anhui Normal University, Anhui 241003, P. R. China e-mail: [email protected] (Received September 25, 2019; revised January 26, 2020; accepted February 5, 2020)
Abstract. We give a regular extension of the Menon-type identity to residually finite Dedekind domains.
1. Introduction Let n be a positive integer. The classical Menon’s identity states that for every n ∈ N, (a − 1, n) = ϕ(n)τ (n), 1 a n (a,n)=1
where ϕ(n) is the Euler’s function and τ (n) = d|n 1. There are many generalizations of Menon’s identity. In a recent paper [19], the authors considered a new generalization of Menon’s identity: gcd(a + b − 1, n) = X(n)τ (n), a,b,a+b∈(Z/nZ)×
where X(n) = a,b,a+b∈(Z/nZ)× 1 and τ (n) is the divisor function. The function X(n) was proposed by Arai and Gakuen [1]. Carlitz [2] showed that ∗ Corresponding
author. This work was partially supported by the Grant No. 11471162 from NNSF of China and the Doctoral Program of QUST (Grant No. 210/010022907). Key words and phrases: arithmetical sum, Menon’s identity, regular map, Dedekind domain, residually finite ring. Mathematics Subject Classification: 11A25, 11B13. c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, Budapest 0236-5294/$20.00 Akade ´miai Kiado ´, Budapest, Hungary
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CH. CH. JI JI and and Y. Y. WANG WANG
X(n) is multiplicative and X(n) = n2
1−
p|n
2 1 1− p p
for every n ∈ N. Actually, we also have X(n) = ϕ(n)2
µ(d) . ϕ(d) d|n
Since the ring of integers OK in a number field K are natural generalizations of the rational integers, the above identity could be deduced for the ring of integers OK in a number field K. Let N (n) be the cardinality of OK /n, ϕ(n) the order of multiplicative group of units in OK /n and τ (n) the number of ideals that divide n. In [19], the authors considered the function X(n) = 1 a,b,a+b∈(OK /n)×
for each ideal n of OK . Then we obtained the generalized Menon-type identity in OK (1) N (�a + b − 1� + n) = X(n)τ (n). a,b,a+b∈(OK /n)×
Recently, Zhao and Cao [21] derived the identity gcd(a − 1, n)χ(a) = ϕ(n)τ (n/d) a∈(Z/nZ)×
with Dirichlet characters, where χ is a Dirichlet character modulo n with conductor d and ϕ(n) is the Euler’s totient function. Zhao and Cao’s paper is not only beautiful but also gives us more information. Following their idea and method, there are many generalizations obtained by several authors, see [5], [6], [7], [15], [16]. In [19], Wang and Ji extended the generalized Menon-type identity (1) in OK with Dirichlet characters, obtained: let χf be a character modulo n with conductor d. Then N (�a + b − 1� + n)χf (a) = µ(d)ϕ(n20/d)X(n/n0)τ (n/d), a,b,a+b∈(OK /n)×
where n0 | n such that n0 has the same prime ideal factors with d and (n0 , nn0 ) = OK . Acta Hungarica Acta Mathematica Mathematica Hungarica
ANOTHER R
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