On the Piltz divisor problem with congruence conditions. II
type: Item
instance of: scholarly article
title: On the Piltz divisor problem with congruence conditions. II
zbMATH Open document ID: 0731.11052
DOI: 10.1007/BF02941055
author: Werner Georg Nowak
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1990-01-01T00:00:00Z
review text: Concerning the first part of this paper [Number theory, Proc. 1st Conf. Can. Number Theory Assoc., Banff/Alberta (Can.) 1988, 455-469 (1990; Zbl 0701.11038)]. For a given \(k\in {\mathbb{N}}\) with \(k=p+q\) (p,q\(\in {\mathbb{N}})\) let \(\ell_ j,m_ j\in {\mathbb{N}}\) with \(\ell_ j<m_ j\) \((j=1,...,p)\). Furthermore let \(d_ k^*(n)\) denote the number of representations of n as \(n=u_ 1\cdot...\cdot u_ k\) satisfying \(u_ j\equiv \ell_ j\) (mod \(m_ j)(j=1,...,p)\). Continuing his studies about \(\Omega\)-results for the error function \[ E_ k(x):=\sum_{n\leq x}d_ k^*(n)- \sum^{k-1}_{i=0}c_ ix(\log x)^ i+O(1);\quad x\to \infty; \] the author now demonstrates - as he announced earlier - for \(k\geq 2\) and \(x\to \infty\) the estimate \[ (*)\quad E_ k(x)=\Omega ((x \log x)^{(k-1)/2k}(\log_ 2x)^{q-1}(\log_ 3x)^{-(k+2)(k-1)/4k}) \] without any restrictions relating the fixed parameters \(p,q,\ell_ j,m_ j\). In case of \(k\geq 4\) this estimate can be refined by replacing \(\Omega_{\pm}\) in (*). Moreover, for \(k\in \{2,3\}\) the author establishes \(\Omega_{\pm}\)-estimates too, but the parameters \(\ell_ j,m_ j\) have to satisfy certain conditions. One of the essential tools in the proof of (*) is a deep result concerning sign changes for certain functions due to \textit{J. Steinig} [J. Number Theory 4, 463-468 (1972; Zbl 0241.10028)].
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11N25
zbMATH DE Number: 4208189
zbMATH Keywords: Piltz divisor problem
zbMATH Keywords: divisors in residue classes
zbMATH Keywords: \(\Omega \) -results
zbMATH Keywords: error function
reviewed by: Winfried Recknagel
MaRDI profile type: Publication
cites work: Q5558293
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cites work: Omega theorems for a class of Dirichlet series
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cites work: Q5803461
full work available at URL: https://doi.org/10.1007/bf02941055
OpenAlex ID: W2089267232
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description: scientific article; zbMATH DE number 4208189