Arithmetical semigroups. IV: Selberg's analysis (Q1325098)

[For Part II, cf. Manuscr. Math. 71, 197-221 (1991; Zbl 0735.11046).] The author develops a subtle counterpart for an arithmetical semigroup \((G,\partial)\) satisfying Axiom \(\text{A}^ \#\) [as defined in the reviewer's monograph ``Analytic arithmetic of algebraic function fields'', M. Dekker (1979; Zbl 0411.10001)] of \textit{A. Selberg's} work on the prime divisor functions \(\omega\) and \(\Omega\) for natural numbers [J. Indian Math. Soc., New Ser. 18, 83-87 (1954; Zbl 0057.285)]. For the parallel functions \(\omega\) and \(\Omega\) on \(G\), he studies the numbers \(\pi_ 1(n,k)\), \(\pi_ 2(n,k)\) and \(\pi_ 3(n,k)\) of elements \(a\in G\) with \(\partial(a)=n\) such that: \(\omega(a)= \Omega(a)=k\), or \(\omega(a)=k\), or \(\Omega(a)=k\), respectively. Delicate asymptotic formulas, too complicated to quote here, are established for these counting numbers when \(1\leq k\leq K\log n\), where \(K>0\) is fixed and arbitrary for the first two cases, but suitably bounded in the last case. [Part V, cf. the paper reviewed below].