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

摘要

B. Sury proved the following Menon-type identity, Sigma gcd(a - 1, b(1), center dot center dot center dot , b(r),n = phi(n)sigma(r)(n), a is an element of U(Z(eta)), b1,center dot center dot center dot ,b(r)is an element of Z(n) where U(Z(n)) is the group of units of the ring for residual classes modulo n, phi is the Euler's totient function and sigma(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 Sigma gcd(a(1) - 1, center dot center dot center dot , a(s) - 1, b(1), center dot center dot center dot , b(r), n) a(1),center dot center dot center dot ,a(s)is an element of U(z(n)), b1,center dot center dot center dot ,b(r)is an element of z(n) m = phi(n) Pi (phi(p(i)(ki))(s-1) p(ki) (r)(i) - p(ki) ((s+r-1))(i) + sigma(s)+r-1 (P-i(ki))), i=1 where n = p(1)(k1) center dot center dot center dot p(m)(km) is the prime factorization of n. (C) 2017 Elsevier Inc. All rights reserved.