On Waring's problem for four cubes
type: Item
instance of: scholarly article
title: On Waring's problem for four cubes
zbMATH Open document ID: 0828.11051
DOI: 10.1215/S0012-7094-95-07718-7
author: Jörg Brüdern
author: Nigel Watt
published in: Duke Mathematical Journal
publication date: 1996-01-07T00:00:00Z
review text: It is generally conjectured that the number of representations of a number \(n\) as a sum of four cubes is of the form \(C_n n^{1/3} + O(E(n))\), where \(C_n = \Gamma^3 ({4 \over 3}) {\mathfrak S} (n)\), \({\mathfrak S} (n)\) being the usual ``singular series'', and \(E(n) = n^{1/3} (\log n)^{- \rho}\) for some constant \(\rho > 0\). \textit{R. C. Vaughan} [J. Reine Angew. Math. 365, 122-170 (1986; Zbl 0574.10046)] showed that this formula holds for all but at most \(O(N (\log N)^{- \rho})\) integers \(n\) not exceeding \(N\). The authors show that the formula holds (with \(\rho = {1 \over 5})\) for all but \(O(M (\log N)^{- 1/4})\) integers in a short interval \(N < n \leq N + M\), where \(M = N^\theta\) and \({5 \over 6} < \theta < 1\). As usual, the Hardy-Littlewood method is used. The usual procedure of invoking a Bessel inequality on the minor arcs would be unsatisfactory in \(M\)-aspect, so the authors introduce a more sophisticated Fourier analysis leading to auxiliary equations in which the \(M\)-dependence is retained.
reviewed by: George Greaves
Mathematics Subject Classification ID: 11P05
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11L15
zbMATH DE Number: 784715
zbMATH Keywords: singular series
zbMATH Keywords: Waring's problem
zbMATH Keywords: number of representations
zbMATH Keywords: sum of four cubes
zbMATH Keywords: Hardy-Littlewood method
Wikidata QID: Q102350766
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784715