Menon-type identities with respect to sets of units (Q2046848)

The classical Menon's identity states that \[ \sum_{a\in\mathbb{Z}_n^\ast}\gcd(a-1, n)=\varphi(n)\tau(n), \] where \(\varphi\) is Euler's totient function and \(\tau\) is the divisor number function, \textit{P. Kesava Menon} [J. Indian Math. Soc., New Ser. 29, 155--163 (1965; Zbl 0144.27706)]. This identity has been generalized in various directions in the literature. The present authors consider a further generalization. Let \(n\ge 1\), \(m\ge 0\), \(k\ge 0\), \(m+k\ge 1\), \(u_1,\dots,u_m\in\mathbb{Z}_n^\star\) and \(S_1,\dots,S_m\) be nonempty subsets of \(\mathbb{Z}_n^\star\). The authors compute the sum \[ \sum_{\substack{a_1\in S_1,\dots,a_m\in S_m \\ b_1,\dots,b_k\in \mathbb{Z}_n}}\gcd(a_1-u_1,\dots,a_m-u_m, b_1,\dots,b_k, n). \] One of the key points here is that summations are over nonempty subsets \(S_i\) of \(\mathbb{Z}_n^\ast\), not necessarily over \(\mathbb{Z}_n^\ast\). Furthermore, the authors compute the sum \(\sum_{t\in S}\gcd(f(t),n)\) for a nonempty subset \(S\) of \(\mathbb{Z}_n^\star\) and any polynomial \(f\) with integer coefficients. This generalizes a well-known Menon-type identity with polynomials, see e.g. [\textit{I. M. Richards}, ``A remark on the number of cyclic subgroups of a finite group'', Am. Math. Mon. 91, No. 9, 571--572 (1984; \url{doi:10.2307/2323746})].