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a\sx absolutely convergent abstract prime number additive arithmetical semigroup algebraic integers algebraic number field analytic function analytic number theory arithmetical category arithmetical functions arithmetical semigroup satisfying asymptotic density asymptotic mean-value C-formula Cg(z Chapter coefficients complex numbers conclusion consider constant coprime Corollary cyclotomic deduce defined degree mapping denote the total Dir(G discussion distribution function divisor divisor function domain elements aŁG elements of G Euclidean domain Euler product formula example finite abelian groups follows formation structure func function on G G satisfies Hence homomorphism implies interesting isomorphism classes Knopfmacher Lemma Let G denote metical multiplicative function NG(x norm number of isomorphism particular PIM-function polynomial positive integers power series prime ideals prime number theorem Proof properties Proposition pseudo-convergent Ramanujan sums real-valued semigroup G semigroup satisfying Axiom sequence shows subset suppose Theorem 3.1 tion topological total number uniformly convergent zeta function