Exponential sums with monomials
type: Item
instance of: scholarly article
title: Exponential sums with monomials
zbMATH Open document ID: 0687.10028
DOI: 10.1016/0022-314X(89)90067-X
author: Étienne Fouvry
author: H. Iwaniec
published in: Journal of Number Theory
publication date: 1989-01-01T00:00:00Z
review text: Exponential sums in several variables \(\sum... \sum e(f(m_ 1,m_ 2,...,m_ j))\), where f is a smooth function, are important in analytic number theory. The paper deals with the estimation of such sums, where f is a monomial function \[ f(m_ 1,m_ 2,...,m_ j)=xm_ 1^{\alpha_ 1}m_ 2^{\alpha_ 2}... m_ j^{\alpha_ j}. \] The method is founded on an inequality for bilinear forms \(\sum \sum \phi_ r\psi_ se(x_ ry_ s)\) with finite sequences of real numbers \(x_ r\) and \(y_ s\) and complex numbers \(\phi_ r\) and \(\psi_ s\), which was proved by \textit{E. Bombieri} and \textit{H. Iwaniec} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 13, 449-472 (1986; Zbl 0615.10047)]. From this inequality it is derived an estimation in case of \(j=4\) and \[ f(m_ 1,...,m_ 4)=x(m_ 1/M_ 1)^{\alpha_ 1}... (m_ 4/M_ 4)^{\alpha_ 4}, \] where the \(m_ i\) in the sum run over intervals of order \(M_ i\). The result is useful in the range \(1\ll x\ll M_ 1... M_ 4\) and will be trivial for larger x. In order to obtain nontrivial results in the latter case the method is combined with Weyl's shifts and Poisson's summation. Finally, applications are given for special sums which occur in sieve problems for short intervals.
reviewed by: Ekkehard Krätzel
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11N35
zbMATH DE Number: 4125487
zbMATH Keywords: Bombieri-Iwaniec inequality
zbMATH Keywords: Exponential sums in several variables
zbMATH Keywords: monomial function
zbMATH Keywords: inequality for bilinear forms
zbMATH Keywords: sieve problems for short intervals
MaRDI profile type: Publication
cites work: Q4723831
cites work: Q3910608
cites work: A new form of the error term in the linear sieve
cites work: \(P_ 2\) in short intervals
cites work: Q5803461
cites work: Q5813998
cites work: Q5760716
Recommended article: Bilinear forms with exponential sums with binomials
Recommended article: Q4397640
Recommended article: Q3337554
Recommended article: Multiple exponential sums with monomials
Recommended article: Double exponential sums and some applications
Recommended article: Exponnential Sums with a Parameter
Recommended article: Trigonometric sums of Heilbronn's type
Recommended article: A special triple exponential sum
Recommended article: Explicit upper bounds for exponential sums over primes
Recommended article: SOME PROBLEMS IN THE INTERFACE BETWEEN NUMBER THEORY, HARMONIC ANALYSIS AND GEOMETRY OF EUCLIDEAN SPACE
description: scientific article; zbMATH DE number 4125487