A Menon-type identity in residually finite Dedekind domains

Let φ   be the Euler's totient function and σk(n)=∑d|ndkσk(n)=∑d|ndk, that is, the sum of the kth powers of the divisors of n. B. Sury showed that∑t1∈U(Zn),t2,…,tr∈Zngcd(t1−1,t2,…,tr,n)=φ(n)σr−1(n), where U(Zn)U(Zn) is the group of units in the ring of residual classes modulo n. Here, this identity is extended to residually finite Dedekind domains.