On characterizations of completely multiplicative arithmetical functions

For an arithmetical function f with f (1) not equal 0 and a positive integer k the generalized von Mangoldt function Lambda(f,k) is defined by [GRAPHICS] where f(-1) is the Dirichlet inverse of f. If f (n) = 1 for all n, then Lambda(f,k) becomes the generalized von Mangoldt function Lambda(k), and if k = 1, then Lambda(k) becomes the usual von Mangoldt function Lambda. Setting the property Lambda(f,k) = f Lambda(k) in the framework of distributivity over the Dirichlet convolution we show that this property serves as a characterization of completely multiplicative functions.