The number of integer points on Vinogradov's quadric
type: Item
instance of: scholarly article
title: The number of integer points on Vinogradov's quadric
zbMATH Open document ID: 1216.11039
author: Valentin Blomer
author: Jörg Brüdern
published in: Monatshefte für Mathematik
publication date: 2010-08-13T00:00:00Z
review text: Let \(V^+(N)\) denote the number of integer solutions to the diophantine system \[ x_1^2+x_2^2+x_3^2=y_1^2+y_2^2+y_3^2,\quad x_1+x_2+x_3=y_1+y_2+y_3 \tag{1} \] with \(1\leq x_j,y_j\leq N\). \textit{N. N. Rogovskaya} [Diophantine approximations, 2, Collect. Artic., Moskva 1986, 78--84 (1986; Zbl 0648.10010)], improving work of Bykovskiĭ, obtained the asymptotic formula \(V^+(N) = \frac{18}{\pi^2} N^3 \log N + O(N^3)\) by an elementary argument. Here, the authors prove a more precise approximation to \(V^+(N)\), namely \[ V^+(N)=\frac{18}{\pi^2}N^3\log N +\frac{3}{\pi^2}\left(12\gamma-6\frac{\zeta'(2)}{\zeta(2)}-5\right)N^3+O(N^{\frac 52}\log N), \] where \(\gamma\) is Euler's constant and \(\zeta\) the Riemann zeta-function. The variety defined by (1) is the simplest one related to Vinogradov's mean value theorem, and therefore the authors refer to it as Vinogradov's quadric. Their motivation to study Vinogradov's quadric arose from two families of paucity theorems. The first of these is connected with Vinogradov's mean value theorem. Alternatively, the quadric (1) arises as the special case \(k = 2\) in the resp. series of pairs of equations.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 5770244
zbMATH Keywords: singular quadric
zbMATH Keywords: integer points
zbMATH Keywords: hyperbola method
zbMATH Keywords: asymptotic estimate
MaRDI profile type: Publication
full work available at URL: s00605-008-0085-8
OpenAlex ID: W1994291511
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DOI: 10.1007/S00605-008-0085-8
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description: scientific article; zbMATH DE number 5770244