Menon-type identities with respect to sets of units

Abstract

Let $n\ge 1$ and let $\mathbb {Z}_n^\star$ be the group of units in the ring of residual classes modulo n, let $m\ge 0$, $k\ge 0$, $m+k\ge 1$, $u_1,\ldots ,u_m\in \mathbb {Z}_n^\star$ and $S_1, \ldots , S_m$ nonempty subsets of $\mathbb {Z}_n^\star$. In this note we shall explicitly compute the following sum $\sum _{\begin{array}{c} a_1\in S_1, \ldots , a_m\in S_m \\ b_1,\ldots , b_k\in \mathbb {Z}_n \end{array}}gcd( a_1-u_1,\ldots , a_m-u_m, b_1,\ldots ,b_k, n).$ Moreover, for a nonempty subset $S\subset \mathbb {Z}_n^\star$ and any polynomial f with integer coefficients we compute the sum $\sum _{t\in S}gcd(f(t), n).$ This generalizes a well-known Menon-type identity with polynomials.