Representation of natural numbers by one cubic form with seven variables
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 \(\varepsilon\).
scientific article; zbMATH DE number 486780