Mean-value theorems of multiplicative functions on additive arithmetic semigroups
type: Item
instance of: scholarly article
title: Mean-value theorems of multiplicative functions on additive arithmetic semigroups
zbMATH Open document ID: 0923.11135
DOI: 10.1007/PL00004658
author: Wenbin Zhang
published in: Mathematische Zeitschrift
publication date: 1999-09-17T00:00:00Z
review text: By an additive arithmetic semigroup it is understood a free commutative semigroup \(G\) with identity element \(1\) such that \(G\) has a countable free generating set \(P\) of ``primes'' \(p\) and such that \(G\) admits an integer-valued degree mapping \(\partial\) which satisfies some suitable conditions. Let \( G(n)\) denote the total number of elements of degree \(n\) in \(G\). In the paper under review the author assumes that \(G(n)\) as \(n\) tends to infinity behaves like \[ G(n)= q^n\sum_{\nu=1}^rA_\nu n^{\rho_\nu-1}+O(q^{n}n^{-\gamma}). \] with some restrictions on \(q,A_\nu,\rho_\nu,\gamma\). He deduces analogs of the well-known Erdős-Wintner theorem and the Halász-Wirsing-Delange theorem, but the exact formulations of his results are too complicate to be given here explicitly. The key result is Theorem 1.1 in which the author considers the asymptotic behavior of the generating function \(\widetilde F(y)\) as \(| y| \to q^{-1}\) depending on that of \(F(m)\) where \(F\) is the summatory function of a multiplicative function \(f\) defined on \(G\). As is well-known, in the classical theory, the values \(f(2^k)\) require special considerations. The same is true in this more general situation, but here the constraints are \(\partial(p)\leq \frac{\log 2}{\log q}\). [Editor's remark: The corrigendum concerns the false theorems 6.4.2 and 1.6.]
reviewed by: Helmut Müller
Mathematics Subject Classification ID: 11N80
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 1268067
zbMATH Keywords: arithmetic semigroups
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: generating function
zbMATH Keywords: summatory function
zbMATH Keywords: multiplicative function
MaRDI profile type: Publication
full work available at URL: pl00004658
OpenAlex ID: W4233660220
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description: scientific article; zbMATH DE number 1268067