A large sieve inequality for rational function fields
type: Item
instance of: scholarly article
title: A large sieve inequality for rational function fields
zbMATH Open document ID: 0852.11049
author: Chih-Nung Hsu
published in: Journal of Number Theory
publication date: 1996-07-01T00:00:00Z
review text: The author derives a large sieve inequality for rational function fields \(\mathbb{F}_q (t)\), where \(\mathbb{F}_q\) denotes a fixed finite field with \(q\) elements. He applies this inequality to obtain an analogue of the Brun-Titchmarsh theorem: Let \(a\), \(b\) be non-zero polynomials in \(\mathbb{F}_q [t]\) with \(a\) monic, \((a, b)= 1\) and \(\deg a>\deg b\geq 0\). For any positive integer \(N\), let \(\pi (N; a, b)\) denote the number of monic irreducibles \(f\in \mathbb{F}_q [t]\) such that \(\deg f= N\) and \(f\equiv b\pmod a\). Then \[ \pi (N; a, b)\leq 2\cdot {{q^N} \over {\Phi (a) (N-\deg a+1)}} \] for \(N> \deg a\), where \(\Phi (a)= \text{card} (\mathbb{F}_q [t]/ (a))^*\). Moreover, the author shows that the set \(\{P\); \(P\), \(P+D\) are monic irreducibles in \(\mathbb{F}_q [t]\}\) has Dirichlet density zero for any non-zero polynomial \(D\in \mathbb{F}_q [t]\).
Mathematics Subject Classification ID: 11N35
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 894607
zbMATH Keywords: large sieve inequality
zbMATH Keywords: rational function fields
zbMATH Keywords: Brun-Titchmarsh theorem
reviewed by: Jürgen Hinz
MaRDI profile type: Publication
full work available at URL: jnth.1996.0077
OpenAlex ID: W2042575534
DOI: 10.1006/JNTH.1996.0077
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description: scientific article; zbMATH DE number 894607