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New omega results in a weighted divisor problem. (English) Zbl 0635.10037
In the paper the weighted divisor sum \(D(a,b;x)=\sum_{m^an^b\leq x}1\), where \(a,b\) are integers with \(1\leq a<b\), is considered. The reviewer [J. Reine Angew. Math. 235, 150–174 (1969; Zbl 0172.05702)] proved that
\[ \Delta (a,b;x)=D(a,b;x)-\zeta (b/a)x^{1/a}-\zeta (a/b)x^{1/b}=\Omega (x^{(a+b)}) \]
and A. Schierwagen [Math. Nachr. 72, 151–168 (1976; Zbl 0285.10030); Publ. Math. 25, 41–46 (1978; Zbl 0392.10039)] gave an improvement of the omega result by applying the methods of G. H. Hardy and E. Landau. The author now proves new omega results.
Schierwagen’s \(\Omega_{+}\)-result is improved in all cases by a logarithm factor. Here the author uses his method developed for Dirichlet’s divisor problem \(a=b=1\) [Invent. Math. 63, 181–186 (1981; Zbl 0458.10031)]. For the proof of the new \(\Omega_{-}\)-result the method of I. Kátai and K. Corrádi [Magyar Tud. Akad., Mat. Fiz. Tud. Oszt. Közl. 17, 89–97 (1967; Zbl 0163.04103)] is used. Since in the problem under consideration there is no functional equation for the generating function an identity for \(D(a,b;x)\), due to the reviewer, is applied in the proofs.

MSC:
11N37 Asymptotic results on arithmetic functions
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