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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
we derive general sufficient conditions for the existence of a limit law for
appropriately normalized multiplicative functions.

Quaestiones Mathematicae
24(3) 2001, 335-347