Summation of values of multiplicative functions on semigroups
Let \(G\) be an arithmetic semigroup with the norm function written additively and satisfying the Axiom \(A^\sharp\) in the notation used by \textit{J. Knopfmacher} [Analytic arithmetic of algebraic function fields (Dekker 1979; Zbl 0411.10001)]. In the paper, a mean value theorem of the Wirsing-Delange-Halász type with a remainder for a class of multiplicative functions defined on \(G\) is proved. Using this result and in a natural way modified notion of the discrepancy over \(G\) a big O and Omega estimation for the discrepancy of the set \(\xi\omega(a)\), where \(\xi\) stands for an irrational number and \(\omega(a)\) for the number of distinct prime divisors of an element \(a\in G\), is proved.
scientific article; zbMATH DE number 794493