On a certain kind of generalized number-theoretical Möbius function (Q2707911)

For each complex number \(\alpha\), the authors define the generalized Möbius function \(\mu_\alpha\) as \(\mu_\alpha(n)=\prod_{p|n} {\alpha\choose\partial_p(n)} (-1)^{\partial_p(n)}\), where \(n=\prod_{p|n} p^{\partial_p(n)}\) is the canonical factorization of \(n\). In particular, \(\mu_1=\mu\), the classical Möbius function. Basic properties and applications of \(\mu_\alpha\) are given.