Über eine Verallgemeinerung des Problems von Chowla and Walum. (On a generalization of the Chowla-Walum conjecture)
Let \(\psi_ k(.)\) denote the 1-periodic Bernoulli polynomial of order \(k\in {\mathbb{N}}\). In generalization of the now well-known conjecture of \textit{S. Chowla} and \textit{H. Walum} [Norske Vid. Selsk. Forh. 36, 127-134 (1963; Zbl 0135.107)] it is natural to conjecture that for \(x\to \infty\) the estimate \[ (*)\quad \sum_{n\leq x^{1/(1+\beta)}}n^{\alpha} \psi_ k(x/n^{\beta})=O(x^{1/(2(1+\beta))+\alpha /(1+\beta)+\epsilon}) \] still holds for reals \(\alpha\geq 0\), \(\beta >0\) and arbitrary \(\epsilon >0\). In case of \(k=1\), \(\alpha =0\) (*) is intimately related to the asymptotic divisor problem in two dimensions. Moreover, for \(k>1\), \(\alpha >\), \(\beta =1\) the conjecture (*) was shown to be true by \textit{S. Kanemitsu} and \textit{R. Sitaramachandrarao} [J. Number Theory 20, 255-261 (1985; Zbl 0467.10031)]; see also the author's paper [Arch. Math. 44, 348-354 (1985; Zbl 0556.10032)]. The objective of the paper under review is to show that (*) holds in case of \(k>1\), \(\alpha >\beta /2\) even with \(\epsilon =0\). In the remaining case \(k>1\), \(\alpha\leq \beta /2\) non-trivial O-results are given.
scientific article; zbMATH DE number 3943946