Representation of natural numbers by one cubic form with seven variables
type: Item
instance of: scholarly article
title: Representation of natural numbers by one cubic form with seven variables
zbMATH Open document ID: 0797.11038
author: Nguyen Khak Tkhan'
published in: Moscow University Mathematics Bulletin
publication date: 1994-04-06T00:00:00Z
review text: In 1943 \textit{Yu. V. Linnik} proved that every large natural number is a sum of seven nonnegative cubes [Dokl. Akad. Nauk SSSR 36, 179-180 (1942)]. Later on Watson gave a different proof of this fact. This settled the question of representability, whereas the question concerning the asymptotic formula for the number of representations remains open. In the present paper we continue studies on this topic [see also the author's earlier paper, Mosc. Univ. Math. Bull. 45, No. 3, 67-68 (1990); translation from Vestn. Mosk. Univ., Ser. I 1990, No. 3, 103-105 (1990; Zbl 0761.11015)]. Let us consider the cubic form \(Q(x,y,z) = x^ 3 + y^ 3 + z^ 3 - 3xyz\). Due to its being decomposable, an asymptotic formula can be derived for the number of representations of the natural number \(N\) by the sum \(Q(x_ 1,x_ 2,x_ 3) + Q(x_ 4,x_ 5,x_ 6) + x^ 3_ 7\). We prove the following: Main theorem. For the number \(J(N)\) of representations of the natural number \(N\) in the form \(N=x^ 3_ 1 + x^ 3_ 2 + \cdots + x^ 3_ 6 + x^ 3_ 7 - 3x_ 1x_ 2x_ 3 - 3x_ 4x_ 5x_ 6\), where \(x_ i\) are integers, \(0 \leq x_ i \leq N^{1/3}\), \(i=1,2,\dots,7\), the asymptotic formula \[ J(N) = \sigma (N) \gamma N^{4/3} + O(N^{152/117 + \varepsilon}) \] holds true, where \(\sigma (N) \geq c>0\), \(\gamma>0\) \((\sigma\) singular series, \(\gamma\) singular integral) and \(\varepsilon > 0\) is an arbitrarily small number. The constant in the symbol ``\(O\)'' depends only on \(\varepsilon\).
Mathematics Subject Classification ID: 11D85
Mathematics Subject Classification ID: 11P55
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 486780
zbMATH Keywords: sum of seven cubes
zbMATH Keywords: asymptotic formula
zbMATH Keywords: number of representations
zbMATH Keywords: cubic form
zbMATH Keywords: singular series
zbMATH Keywords: singular integral
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 486780