The abstract prime number theorem for function fields
type: Item
instance of: scholarly article
title: The abstract prime number theorem for function fields
zbMATH Open document ID: 0789.11056
DOI: 10.1007/BF01874225
published in: Acta Mathematica Hungarica
publication date: 1994-06-13T00:00:00Z
review text: On the basis mainly of explicit motivating examples arising algebraically in relation to function fields over a finite field, the reviewer [Analytic Arithmetic of Algebraic Function Fields (M. Dekker, New York, 1979; Zbl 0441.10001)] investigated additive arithmetical semigroups \(G\) satisfying the Axiom \(A^ \# :G^ \# (n)=Aq^ n+O (q^{\nu n})\) for constants \(A>0\), \(q>1\), \(0 \leq \nu<1\), where \(G^ \# (n)\) denotes the number of elements of degree \(n\) in \(G\). It is of special interest in this connection to determine asymptotic properties of the number \(P^ \# (n)\) of prime elements of degree \(n\) in \(G\). This question and others are related to properties of \(Z_ G^ \# (y):=\sum^ \infty_{n=0}G^ \# (n)y^ n\), and in partially answering it under Axiom \(A^ \#\) the reviewer overlooked the need for a special assumption: \(Z^ \#_ G(- q^{-1}) \neq 0\). This latter condition is valid for all the initial motivating examples and therefore in principle could have been covered by an innocuous additional assumption or axiom. Nevertheless K.-H. Indlekofer, R. Warlimont and others showed about 14 years later that the Axiom \(A^ \#\) alone does not imply \(Z_ G^ \# (-q^{-1})\neq 0\). They then successfully investigated \(P^ \# (n)\) when \(Z_ G^ \# (- q^{-1})=0\); [see for example \textit{K.-H. Indlekofer}, \textit{E. Manstavičius} and \textit{R. Warlimont}, Arch. Math. 56, 446-453 (1991; Zbl 0708.11041)]. The main thrust of the present interesting paper is to investigate sharper formulae for \(P^ \# (n)\) under a new Axiom \(\overline A^ \#\) about analytical properties of the function \(Z(y):=(1-qy)Z^ \#_ G(y)\): It is assumed that there exist constants \(q>1\) and \(0 \leq \nu<1\) such that: (i) \(Z(y)\) is holomorphic for \(| y |<q^{-\nu}\), with \(Z(q^{-1})>0\), and (ii) as a function of \(z\), \(Z(q^{-\nu}z)\) is an element of the Nevanlinna class \(N\). Unlike the case of Axiom \(A^ \#\) together with \(Z^ \#_ G(-q^{- 1})=0\), for which no interesting non-pat…
Mathematics Subject Classification ID: 11N80
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 480505
zbMATH Keywords: analytic arithmetic of algebraic function fields
zbMATH Keywords: number of prime elements
author: K.-H. Indlekofer
reviewed by: John Knopfmacher
MaRDI profile type: Publication
cites work: The function field abstract prime number theorem
cites work: Q5531597
cites work: On the abstract theory of primes I
cites work: Q4083499
cites work: An abstract prime number theorem relating to algebraic function fields
cites work: Q4405392
full work available at URL: https://doi.org/10.1007/bf01874225
OpenAlex ID: W1968069005
Recommended article: The function field abstract prime number theorem
Recommended article: Q3211412
Recommended article: The prime element theorem in additive arithmetic semigroups. I
Recommended article: Q4198823
Recommended article: Q4660236
Recommended article: Remarks on the number theory over additive arithmetical semigroups
Recommended article: New approach to multiplicative functions on arithmetical semigroups
Recommended article: Tauberian Theorems for Sequences linked by a Convolution
Recommended article: Q4684623
Recommended article: Q4849855
description: scientific article; zbMATH DE number 480505