Une conjecture de Chowla et Walum. (A conjecture of Chowla and Walum)
type: Item
instance of: scholarly article
title: Une conjecture de Chowla et Walum. (A conjecture of Chowla and Walum)
zbMATH Open document ID: 0568.10023
DOI: 10.1016/0022-314X(85)90053-8
author: Jacques Meyer
published in: Journal of Number Theory
publication date: 1985-01-01T00:00:00Z
review text: 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)].)
Mathematics Subject Classification ID: 11N37
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 3906587
zbMATH Keywords: divisor problems
zbMATH Keywords: Bernoulli functions
zbMATH Keywords: Chowla-Walum conjecture
Wikidata QID: Q122955364
MaRDI profile type: Publication
cites work: Q5507778
cites work: On the order of zeta(1/2+it) and Delta(R)
cites work: Q5666618
description: scientific article; zbMATH DE number 3906587