Fermat and the quality of Riemann approximations for \(\int_{-1}^1\int_{-1}^1 \int_{-1}^1 \varPhi ((ax+by+cz)R)\varPsi(x,y,z)\,dx\,dy\,dz\)
type: Item
instance of: scholarly article
title: Fermat and the quality of Riemann approximations for \(\int_{-1}^1\int_{-1}^1 \int_{-1}^1 \varPhi ((ax+by+cz)R)\varPsi(x,y,z)\,dx\,dy\,dz\)
zbMATH Open document ID: 1209.11090
author: Klaus Langmann
published in: Journal of Number Theory
publication date: 2008-11-18T00:00:00Z
review text: Given an even fast-decreasing \(C^{\infty}(\mathbb{R})\)-function \(\Phi\), and a holomorphic function \(\Psi\) on the polycylinder \(D(0,1+\theta)^3\), the author considers the integral \(I(\Phi,\Psi) = \int_{-1}^1 \int_{-1}^1\int_{-1}^1 \Phi( (ax+by+cz) R) \Psi(x,y,z)\,dx \,dy\,dz\) for fixed integers \(a,b,c\) in comparison with the Riemann sum \[ \begin{multlined} S(\Phi, \Psi, \xi) = \\ \sum_{-R < i,j,k \leq R} \Phi ((a \xi_i + b\xi_j +c \xi_k )R ) \Psi (\xi_i, \xi_j, \xi_k) (\xi_i-\xi_{i-1}) (\xi_j-\xi_{j-1}) (\xi_k-\xi_{k-1}),\end{multlined} \] where the evaluation points take the polynomial value \(\xi_i = (i/R)^n\) for a given \(n\). As a consequence of his work, he is able to show that for \(n\) even and \(n \geq 5\), the error term is of the order of \(O(R^{-3/2})\) if the equation \(ax^n + by^n + cz^n = 0\) has a non trivial solution and \(O(R^{-2 + \varepsilon})\) otherwise.
reviewed by: Bertrand Meyer
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11D41
zbMATH DE Number: 5368226
zbMATH Keywords: Riemann sum approximation
zbMATH Keywords: Fermat equation
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.05.001
OpenAlex ID: W18581161
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Unsolvability of the equation \(\alpha x^n + \beta y^n = z^n\) and uniform distribution of \(\root n \of {\alpha x^n + \beta y^n}\)
cites work: Q4010347
DOI: 10.1016/J.JNT.2008.05.001
description: scientific article; zbMATH DE number 5368226