Estimates of Weyl's exponential sums for very small values of the variable
type: Item
instance of: scholarly article
title: Estimates of Weyl's exponential sums for very small values of the variable
zbMATH Open document ID: 1083.11053
author: Ekkehard Krätzel
published in: Archiv der Mathematik
publication date: 2005-01-20T00:00:00Z
review text: Let \(k, N\) be integers, \(k\geq 3\) and let \[ p_k (z) = a_1 z + \dfrac{a_2} 2 z^2 + \cdots + \dfrac{a_k} k z^k \] be a polynomial of degree \(k\) with coefficients \(a_i \in \mathbb R\), and \(a_k > 0\). The author considers in this article the general Weyl sum \[ S= \sum\limits^N_{n=1} e^{2\pi i \vartheta p_k (n)} \] where, as usual, \(0 < \vartheta <1 , \quad \vartheta N^k > 1\). This is an extension of previous works of the author for the case \(p_k (z) = z^k\). The main result obtained is a transformation formula of the form \(S=F + E + {\mathcal O} (\sqrt N)\) where \(F\) is the main term and \(E\) is another exponential sum. Then by using van der Corput's method of estimating exponential sums, \(E\) is bounded satisfactorily over the range \(1/p'_k (N) < \theta < N^{-k+ 2 + \delta}\) for a positive number \(\delta \leq 1\). The main feature here is that the range for \(\vartheta\) extends beyond \(N^{2-k}\). The detailed results are too complicated to be reproduced here.
Mathematics Subject Classification ID: 11L15
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 2130542
zbMATH Keywords: Weyl sums
zbMATH Keywords: exponential sums
zbMATH Keywords: van der Corput's method
reviewed by: Kai-Man Tsang
MaRDI profile type: Publication
full work available at URL: s00013-004-4656-6
OpenAlex ID: W151994775
DOI: 10.1007/S00013-004-4656-6
description: scientific article; zbMATH DE number 2130542