Competing prime asymptotic densities in \(\mathbb F_q[X]\): a discussion
type: Item
instance of: scholarly article
title: Competing prime asymptotic densities in \(\mathbb F_q[X]\): a discussion
zbMATH Open document ID: 1239.11011
author: Christian Ballot
published in: L'Enseignement Mathématique. 2e Série
publication date: 2009-02-26T00:00:00Z
review text: In the paper under review, the author introduces four densities on the set of prime (irreducible) polynomials over \(\mathbb F_q[X]\). For a positive integer \(n\) let \(I_n\) be the number of prime polynomials of degree \(n\) in \(\mathbb F_q[X]\). Given a set \(S\) of polynomials in \(\mathbb F_q[X]\), let \(S_n\) be the number of prime polynomials of degree \(n\) in \(S\). Definition 1.3 says that \(S\) has density \(d_1,~d_2,~d_3,\) and \(d_4\) if the sequence of general term \[ {{S_n}\over {I_n}},\quad {{\sum_{k\leq n} S_k}\over {\sum_{k\leq n} I_k}},\quad {{1}\over {n}}\sum_{k\leq n} {{S_k}\over {I_k}}\quad {\text{and}}\quad {{\sum_{k\leq n} S_k/q^k}\over {\sum_{k\leq n} I_k/q^k}}, \] respectively, has a limit. In case the corresponding limit exists it is denoted by \(d_i(S)\) for \(i=1,\dots,4\). The main result of the paper is summarized as Theorem A of Section 4. For example, \(d_1\) exists if and only if \(d_2\) exists and whenever they exist for some \(S\), then \(d_1(S)=d_2(S)\). The existence of (either one) of \(d_1(S)\) or \(d_2(S)\) implies the existence of \(d_3(S)\), and the existence of \(d_3(S)\) implies the existence of \(d_4(S)\), which in turn is equivalent to the Dirichlet density. The paper is devoted to the proof of these implications and abounds with examples and an enlightening discussions as to the advantages and disadvantages offered by working with each of the five densities (the four defined above and the Dirichlet density).
reviewed by: Florian Luca
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11T55
zbMATH DE Number: 5519678
zbMATH Keywords: irreducible polynomials modulo \(p\)
zbMATH Keywords: relative densities of primes in \(F_q[X]\)
MaRDI profile type: Publication
DOI: 10.5169/SEALS-109941
description: scientific article; zbMATH DE number 5519678