On characterizations of completely multiplicative arithmetical functions (Q2710118)

For an (arithmetical) function \(f: \mathbb{N}\to \mathbb{C}\) and a positive integer \(k\) the generalized von Mangoldt function is defined by \(\Lambda_{f,k}= f^{-1}* (f\log^k)\), where \(f*g\) is the Dirichlet convolution and \(f^{-1}\) the Dirichlet inverse of \(f\). The author proves that an arithmetical function \(f\) with \(f(1)=1\) is completely multiplicative if and only if \(\Lambda_{f,k}= f\Lambda_{u,k}\), where \(u\) is the constant function 1, so \(\Lambda_{u,k}= \mu* \log^k\). The proof is in the framework of distributivity over the Dirichlet convolution.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].