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On the Piltz divisor problem with congruence conditions. II. (English) Zbl 0731.11052
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 J. Steinig [J. Number Theory 4, 463-468 (1972; Zbl 0241.10028)].

MSC:
11N37 Asymptotic results on arithmetic functions
11P21 Lattice points in specified regions
11N25 Distribution of integers with specified multiplicative constraints
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