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

B. Sury proved the following Menon-type identity, ∑a∈U(Z_n),b₁,⋯,b_r∈Z_ngcd⁡(a−1,b₁,⋯,b_r,n)=φ(n)σ_r(n), where U(Z_n) is the group of units of the ring for residual classes modulo n, φ is the Euler's totient function and σ_r(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,⋯,a_s∈U(Z_n),b₁,⋯,b_r∈Z_ngcd⁡(a₁−1,⋯,a_s−1,b₁,⋯,b_r,n)=φ(n)∏i=1m(φ(p_i^k_i)^s−1p_i^k_ir−p_i^k_i(s+r−1)+σ_s+r−1(p_i^k_i)), where n=p₁^k₁⋯p_m^k_m is the prime factorization of n.