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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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