On Gauss sums and the evaluation of Stechkin's constant
type: Item
instance of: scholarly article
zbMATH Open document ID: 1339.11078
author: William D. Banks
author: Igor E. Shparlinski
published in: Mathematics of Computation
publication date: 2016-06-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.5669
Mathematics Subject Classification ID: 11L05
zbMATH DE Number: 6596199
zbMATH Keywords: Stechkin's constant
zbMATH Keywords: sphere packing
zbMATH Keywords: Gauss sums
MaRDI profile type: Publication
OpenAlex ID: W2073962687
arXiv ID: 1310.5669
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DOI: 10.1090/MCOM3056
title: On Gauss sums and the evaluation of Stechkin's constant
review text: The Gauss sums are defined by NEWLINE\[NEWLINE S_n\left(a,q\right)=\sum_{x\bmod q}\text{e}\left(\frac{ax^n}{q}\right), NEWLINE\]NEWLINE where \(\text{e}(t)=e^{2\pi it}\) for all \(t\in \mathbb{R}\). Denote NEWLINE\[NEWLINE A(n)=\sup_{q\geq 2}\max_{\text{gcd}(a,q)=1}\frac{\left|S_n\left(a,q\right)\right|}{q^{1-\frac{1}{n}}}. NEWLINE\]NEWLINE This paper shows that \(A(n)<A(6)\) for all \(n\geq 2\), \(n\neq 6\), and NEWLINE\[NEWLINE A(6)=\frac{\left|S_6\left(4787, 4606056\right)\right|}{4606056^{\frac{5}{6}}}=4.709236\dotsNEWLINE\]
description: scientific article; zbMATH DE number 6596199