An analogue to a conjecture of S. Chowla and H. Walum
type: Item
instance of: scholarly article
title: An analogue to a conjecture of S. Chowla and H. Walum
zbMATH Open document ID: 0546.10039
DOI: 10.1016/0022-314X(84)90110-0
author: Werner Georg Nowak
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: The author considers an analogue, in the case of the Piltz divisor problem for three dimensions, to a conjecture of Chowla and Walum which itself corresponds to the Dirichlet divisor problem, and proves the estimate \[ (*)\quad G_{a,b;k}(x):=\sum_{n\leq x^{1/3}}\sum_{n<m\leq (x/n)^{1/2}}n^ a m^ b B_ k(\{x/nm\})=O(x^{(a+b+1)/3+\epsilon}) \] for every \(\epsilon >0\), where \(\{\) \(a\}\) denotes the fractional part of \(\alpha\), and \(B_ k(x)\) denotes the k-th Bernoulli polynomial. Generalization of (*) to higher dimension would be interesting, but the author states that it appears hopeless as far as no counterpart of Vinogradov's shifting interval lemma is known.
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 3869449
zbMATH Keywords: Chowla-Walum conjecture
zbMATH Keywords: Piltz divisor problem
zbMATH Keywords: Dirichlet divisor problem
zbMATH Keywords: Bernoulli polynomial
Wikidata QID: Q123241780
reviewed by: Shigeru Kanemitsu
MaRDI profile type: Publication
cites work: Q5507778
cites work: Q3950619
cites work: ON A KIND OF DIVISOR PROBLEM
cites work: On a kind of divisor problem. II
cites work: On a conjecture of S. Chowla and of S. Chowla and H. Walum. III
cites work: On the order of zeta(1/2+it) and Delta(R)
cites work: Q5803461
cites work: Q3866177
full work available at URL: 0022-314x(84)90110-0
OpenAlex ID: W1975864831
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description: scientific article; zbMATH DE number 3869449