Arithmetical semigroups. III: Elements with prime factors in residue classes
The author studies an arithmetical semigroup \((G, \partial)\) satisfying Axiom \(A^\#\) as defined and stated in the reviewer's monograph [Analytic arithmetic of algebraic function fields, M. Dekker (1979; Zbl 0411.10001)]. The asymptotic distribution of elements of \(G\) is investigated subject to the condition that the degrees \(\partial (p)\) of the prime elements \(p\) of the canonical prime decompositions all lie in a given union of integer residue classes modulo a fixed integer \(k\). The results obtained are analogous to those for the classical similar problem concerning natural numbers, derived in Section 183 of \textit{E. Landau} [Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea (1953; Zbl 0051.280)]. Nevertheless the analogy is not trivial, involving different types of formulae and analytical techniques. [Further parts of this series have been reviewed in (I) Zbl 0723.11048, (II) Zbl 0735.11046, (IV) Zbl 0809.11058, (V) Zbl 0809.11059].
scientific article; zbMATH DE number 742705