Une conjecture de Chowla et Walum. (A conjecture of Chowla and Walum)
The conjecture of S. Chowla and H. Walum referred to in the title states that, for arbitrary integers \(k\geq 2\) and reals \(a>0\), \[ \sum_{n\leq \sqrt{x}}n^ a B_ k(x/n)=O(x^{a/2+1/4}) \] where \(B_ k( )\) is the periodic Bernoulli function of order k [see \textit{S. Chowla} and \textit{H. Walum}, Norske Vid. Selsk. Forh. 36, 127-134 (1963; Zbl 0135.107)]. This was shown to be true for \(a>\) by \textit{S. Kanemitsu} and \textit{R. Sitaramachandra Rao} [J. Number Theory 20, 255-261 (1985; Zbl 0467.10031)] by a very short and elegant proof involving the Fourier series for \(B_ k( )\) and Van der Corput's method to estimate exponential sums. In the paper under review, the above assertion is established for the special case \(a+1=k\geq 2\) only, by a rather artificial and complicated argument. (For the sharpest result to date on the remaining case \(0\leq a\leq\), see \textit{W. Recknagel} [Arch. Math. 44, 348-354 (1985; Zbl 0556.10032)].)
scientific article; zbMATH DE number 3906587