A Menon-type identity with many tuples of group of units in residually finite Dedekind domains

Abstract

B. Sury proved the following Menon-type identity,∑a∈U(Zn),b1,⋯,br∈Zngcd⁡(a−1,b1,⋯,br,n)=φ(n)σr(n), where U(Zn) is the group of units of the ring for residual classes modulo n, φ is the Euler's totient function and σr(n) is the sum of r-th powers of positive divisors of n with r being a non-negative integer. Recently, C. Miguel extended this identity from Z to any residually finite Dedekind domain. In this note, we will give a further extension of Miguel's result to the case with many tuples of group of units. For the case of Z, our result reads as follows∑a1,⋯,as∈U(Zn),b1,⋯,br∈Zngcd⁡(a1−1,⋯,as−1,b1,⋯,br,n)=φ(n)∏i=1m(φ(piki)s−1pikir−piki(s+r−1)+σs+r−1(piki)), where n=p1k1⋯pmkm is the prime factorization of n.