On \(s\)-dimensional incomplete Kloosterman sums
type: Item
instance of: scholarly article
title: On \(s\)-dimensional incomplete Kloosterman sums
zbMATH Open document ID: 1208.11096
author: Yunjie Wang
author: Hongze Li
published in: Journal of Number Theory
publication date: 2010-07-20T00:00:00Z
review text: Let \[ K_{s}(\vec{a},\vec{M},\vec{N};p)=\sum\limits_{n_{1}=M_{1}+1}^{M_{1}+N_{1}}\cdots\sum\limits_{n_{s}=M_{s}+1}^{M_{s}+N_{s}}e_{p}(a_{1}n_{1}+\cdots+a_{s}n_{s}+a_{s+1}\overline{n_{1}\cdots n_{s}}), \] where \(\vec{a}=(a_{1},\dots,a_{s+1})\in\mathbb{Z}^{s+1}\), \(\vec{M}=(M_{1},\dots,M_{s})\in\mathbb{Z}^{s}\), \(\vec{N}=(N_{1},\dots,N_{s})\in\mathbb{Z}^{s}\), \(0\leq M_{v}<M_{v}+N_{v}<p\), \(v=1,2,\dots,s \) and \(p\) is a prime. The authors utilize the estimation of number of solutions of congruence to obtain : \[ K_{s}(\vec{a},\vec{M},\vec{N};p)\ll \frac{N_{1}\cdots N_{s}}{p}+(N_{1}\cdots N_{s})p^{-1/2+\varepsilon}\prod\limits_{v=1}^{s} (1+N_{v}^{-1/2}p^{1/s}),\;s=2k\geq 4;\tag{i} \] \[ K_{s}(\vec{a},\vec{M},\vec{N};p)\ll \frac{N_{1}\cdots N_{s}}{p}+(N_{1}\cdots N_{s})^{1-1/2s}p^{\varepsilon}\prod\limits_{v=1}^{s} (1+N_{v}^{\frac{1-s}{2s}}p^{1/2s}),\;s=2k+1\geq 3.\tag{ii} \]
reviewed by: Zaizhao Meng
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11A07
zbMATH DE Number: 5758005
zbMATH Keywords: Kloosterman sums
zbMATH Keywords: congruences
zbMATH Keywords: solutions
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.02.013
OpenAlex ID: W2032793967
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DOI: 10.1016/J.JNT.2010.02.013
description: scientific article; zbMATH DE number 5758005