On an estimate of the Vinogradov integral for a small number of variables
In the paper two estimates of the Vinogradov integral [\textit{I. M. Vinogradov}, ``Method of trigonometric sums in the theory of numbers'', Nauka, Moscow (1980; Zbl 0561.10017), reprint of the English translation Dover Publications (2004; Zbl 1093.11001)] \[ J_{k,n}(P) = \int\limits_0^1\dots\int\limits_0^1 \Bigg| \sum\limits_{x=1}^P \exp2\pi if(x)\Bigg| ^{2k} d\alpha_1\dots d\alpha_n \] where \(f(x) = \alpha_1x+\dots+\alpha_n x^n\), representing the number of solutions of the system of equations \[ x^j_1+\dots+x^j_n=y^j_1+\dots+y^j_n \qquad (1\leq j\leq k) \qquad x_1,y_1,\dots,x_n,y_n\in\{1,\dots,P\} \] are presented. The author assumes \(P>1\), \(k\geq n\geq 6\) and considers two cases: (1) \(m=k-n\leq(n+3)/2\), (2) \(k=\delta n^2\leq 3n^2/4\), and obtains estimates of the form \(J_{k,n}(P)\leq C_1P^{k+c}\). In the first case one has \(C_1=n^{4.8n^2}\), \(c=4m(n+1)/(n+2)^2\) for even \(n\), \(c=4m/(n+3)\) for odd \(n\). In the second case one has \(C_1=n^{3.5n^2}\), \(c=(\delta-1/n)(8k-3.6n+0.72)\).
scientific article; zbMATH DE number 1977027