Fermat and the quality of Riemann approximations for \(\int_{-1}^1\int_{-1}^1 \int_{-1}^1 \varPhi ((ax+by+cz)R)\varPsi(x,y,z)\,dx\,dy\,dz\)
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.
scientific article; zbMATH DE number 5368226