Über eine Vermutung von S. Chowla und H. Walum
Let \(\psi_k(\cdot)\) denote the 1-periodic Bernoulli polynomial of order \(k\in\mathbb{N}\). In generalization of Dirichlet's classical divisor problem, \textit{S. Chowla} and \textit{H. Walum} [Norske Vid. Selsk. Forh. 36, 127--134 (1963; Zbl 0135.10703)] conjectured that for \(x\to \infty\) the estimate \[ \sum_{n\leq x^{1/2}}n^{\alpha}\psi_ k(x/n) = O(x^{\vartheta (\alpha)});\ \vartheta (\alpha)=1/4+\alpha /2 + \varepsilon \tag{*} \] still holds for real \(\alpha\geq 0\) and arbitrary \(\varepsilon >0\). \textit{S. Kanemitsu} and \textit{R. Sitaramachandrarao} [J. Number Theory 20, 255--261 (1985; Zbl 0467.10031)] established the conjecture in case of \(k\geq 2\) and \(\alpha\geq 1/2\). \textit{W. G. Nowak} [Bull. Number Theory Rel. Top. 7, No. 1, 1--10 (1982; Zbl 0491.10013)] improved their results in the remaining case \(k\geq 2\) and \(0\leq \alpha <1/2\), e.g. \(\vartheta (0)=2/7=0,2857\ldots\). The objective of the paper is to present a uniform approach for estimating sums like (*) in the case \(k\geq 2\). The corresponding exponential sum is estimated with the aid of van der Corput's theory of exponent pairs in the simplified form by \textit{E. Phillips} [Q. J. Math., Oxf. Ser. 4, 209--225 (1933; Zbl 0007.29801)] using an argument of \textit{H.-E. Richert} [Math. Z. 56, 21--32 (1952; Zbl 0046.25002)]. As an application of the theorem one gets slightly refinements over the results of W. G. Nowak (loc. cit.), e.g. \(\vartheta (0)=109/382=0,2853\ldots\). For \(k\geq 2\), \(\alpha\geq 1/2\) the conjecture is proved within a few lines. \{Note: W. G. Nowak pointed out to the author that the special case \(k=2\), \(\alpha =0\) in (*) is implicitly Wigert's divisor problem. The best known result about the corresponding exponent hitherto is that of \textit{H. Y. Peng} [Sci. Record 1, 69--72 (1942; Zbl 0060.10406)] namely \(2/7\) and can be replaced by \(109/382\).\}
scientific article; zbMATH DE number 3886007