Some number-theoretic identities from group actions.
Group actions are applied to some natural situations like the natural `linear' action of \(\text{GL}_r(\mathbb{Z}_n)\) and some of its subgroups to derive number-theoretic identities like \[ \sum_{t_1\in(\mathbb{Z}_n)^*,t_2,\dots,t_r\in\mathbb{Z}_n}\text{GCD}(n,t_1-1,t_2,\dots,t_r)=\varphi(n)\sigma_{r-1}(n). \] Note, that the case \(r=1\) yields the identity \[ \sum_{\text{``\(a\)''}}\text{GCD}(a-1,n)= \varphi(n)d(n), \] where \(d(n)\) stands for the divisor function of \(n\), and where \(\displaystyle\sum_{\text{``\(a\)''}} = \sum_{\text{GCD}(a,n)=1}\).
scientific article; zbMATH DE number 5620617