Publ. Math. Debrecen 45 / 1-2 (1994), 1–17 Additive and multiplicative functions on arithmetical semigroups By KARL-HEINZ INDLEKOFER (Paderborn) and EUGENIJUS MANSTAVI ˇ CIUS (Vilnius) 1. Introduction Many problems concerning additive and multiplicative functions de- fined on N can be posed and solved in a more abstract setting. One can investigate functions on an arithmetical semigroup which by definition is a commutative semigroup G with identity element 1, and which contains a countable subset P such that every element a = 1 in G admits unique factorization into a finite product of powers of elements of P . The direct generalization of N is the arithmetical semigroup satisfying Axiom A. A completely multiplicative norm function ‖·‖ is defined on G so that ‖p‖ > 1 for each p ∈P , and there exist constants A> 0, 0 ≤ θ ′ < 0 such that (1) #{a ∈G ; ‖a‖≤ x} = Ax θ + O(x θ ′ ). The development of analytic and probabilistic number theory in such semigroups is represented by J. Knopfmacher’s monograph [8], papers quoted in it and more recent publications. The semigroup of primary polynomials over a finite field as well as that of the integral divisors in al- gebraic function fields and many other interesting arithmetical semigroups do not fall under the scope of Axiom A because the regularity of norms of elements has different character. These semigroups satisfy Axiom A ∗ . A completely additive degree function ∂ is defined on G so that ∂ (p) ≥ 1 for each p ∈P and G (n) := #{a ∈G ; ∂ (a)= n} = Aq n + O(q νn ) This work was done while the second author held a visiting professorship at the Pader- born Universit¨ at supported by the Deutsche Forschungsgemeinschaft. Typeset by A M S-T E X