Sums of moduli of exponential sums
type: Item
instance of: scholarly article
title: Sums of moduli of exponential sums
zbMATH Open document ID: 1061.11044
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: Let \(Kl(a,b; n)\) be the usual Kloosterman sum, and let \[ A^*(x)= \sum_{n\leq x} {|Kl(1,1;n)|\over 2^{\omega(n)}\sqrt{n}},\;\widetilde A(x)= \sum_{n\leq x} {|Kl(1,1;n)|\over \sqrt{n}}. \] It is shown that there is a constant \(\delta> 5/12\) such that \[ {x\over\log x} \exp((\log\log x)^\delta)\ll A^*(x)\ll x\Biggl({\log\log x\over\log x}\Biggr)^{1-4/3\pi} \] and similarly \[ {x\over\log x}\exp((\log\log x)^\delta)\ll\widetilde A(x)\ll x\Biggl({\log\log x\over\log x}\Biggr)^{1-8/3\pi}. \] Such estimates were first considered by \textit{C. Hooley} [Mathematika, Lond. 11, 39--49 (1964; Zbl 0123.25802)] forty years ago, who gave, in effect, a bound \[ \widetilde A(x)\ll x(\log x)^{\sqrt{2}-1}(\log\log x)^c. \] Analogous results for certain more general exponential sums in one variable are also presented. The proof is a technical masterpiece, combining a result of the second author [Duke Math. J. 95, No. 2, 227--240 (1998; Zbl 0958.11056)] on the distribution of exponential sums, with a large sieve inequality, and an analysis of the contributions from ``small'' and ``large'' prime factors.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 2101073
zbMATH Keywords: Kloosterman sums
zbMATH Keywords: absolute value
zbMATH Keywords: mean value
zbMATH Keywords: exponential sums
author: Étienne Fouvry
author: Philippe Michel
MaRDI profile type: Publication
full work available at URL: pjm.2003.209.261
OpenAlex ID: W1973039890
DOI: 10.2140/PJM.2003.209.261
description: scientific article; zbMATH DE number 2101073