DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS DEFINED ON SEMIGROUPS

Abstract

The value distribution problem for real-valued multiplicative functions defined on an additive arithmetical semigroup is examined. We prove that, in contrast to the classical theory of number-theoretic functions defined on the semigroup of natural numbers, this problem is equivalent to that for additive functions only under some extra condition. In this way, applying the known results for additive functions we derive general sufficient conditions for the existence of a limit law for appropriately normalized multiplicative functions.