Semimultiplicative generalized arithmetical functions

Abstract

By a generalized arithmetical function we mean a function from the set of positive integers to a ring with identity, and we say that a generalized arithmetical function f is semimultiplicative if f(n) = cf fM (n/af), where cf is a unit in the ring, af is a positive integer and fM is a multiplicative generalized arithmetical function. We study basic properties of these functions, connections to Selberg multiplicative functions and to the Dirichlet convolution. Particular attention is paid to the commutativity and noncommutativity of the function values.