The abstract prime number theorem for function fields
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-pathological examples seem to be known, the author strongly motivates his new axiom by stating that it is satisfied by all the earlier concrete examples, and he establishes this claim in detail for a large proportion of them.
analytic arithmetic of algebraic function fields
scientific article; zbMATH DE number 480505