Multiplicative character sums and products of sparse integers in residue classes
type: Item
instance of: scholarly article
title: Multiplicative character sums and products of sparse integers in residue classes
zbMATH Open document ID: 1263.11077
author: Alina Ostafe
author: Igor E. Shparlinski
published in: Periodica Mathematica Hungarica
publication date: 2012-09-17T00:00:00Z
full work available at URL: ZORA NL 156024.pdf
review text: Let \( n=\sum_{j\geq0}a_{j}(n)2^{j} \), with \( a_{j}(n)=0, 1 \) be the binary representation of \(n\) and let \( \mathcal{G}_{s}(r) \) be the set of integers with \(r\) binary integers such that the sum of the digits is equal to \(s\). The authors of this paper discuss character sums of the form \(S_{s}(r,\chi,f)\), where \(\chi\) is a non-principal multiplicative character of the finite field \( \mathbb{R}_{p} \) with \(p\) elements, and \(f(X)\) is a polynomial in \( \mathbb{R}_{p} \). They prove three theorems. One of these is: Theorem 2. For any \( \gamma > \rho_{0}=0.11002786... \), there exists some \( \eta > 0 \) such that if \( p>2^{r}=p^{1+o(1)} \) and \(r/2\geq s\geq \gamma r\), then for any nontrivial multiplicative character \(\chi\) modulo \(p\), we have \(S_{s}(r,\chi, f) \ll {r \choose s}^{1-\eta}\).
reviewed by: Tonko Tonkov
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 6083230
zbMATH Keywords: multiplicative character sums
zbMATH Keywords: congruences
zbMATH Keywords: sparse integers
zbMATH Keywords: binary digits
MaRDI profile type: Publication
OpenAlex ID: W2081459967
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DOI: 10.1007/S10998-012-6771-2
description: scientific article; zbMATH DE number 6083230