Characterizations of L-additive functions via generalized arithmetic convolutions

This paper investigates the properties of L-additive functions within the algebraic frameworks of two generalized arithmetic convolutions: the K-convolution and Narkiewicz’s A-convolution. We establish the foundational algebraic context for these operations by citing the established conditions for their associativity and commutativity. Our main results provide rigorous characterization theorems for completely additive and L-additive functions, which manifest as Leibniz-type rules that these functions satisfy with respect to the convolutions. Furthermore, we provide insightful, non-trivial examples using classical arithmetic functions to illustrate the mechanics of these characterizations, thereby demonstrating the utility of the generalized convolution framework in the study of arithmetic derivatives and their generalizations.