The prime element theorem in additive arithmetic semigroups. I (Q1922057)

The abstract prime number theorem for an algebraic function field is usually proved in the context of an additive arithmetic semigroup since the concept of the latter was introduced by \textit{J. Knopfmacher} [Analytic arithmetic of algebraic function fields, Marcel Dekker, New York (1979; Zbl 0411.10001)]. By an additive arithmetic semigroup it is understood a free commutative semigroup \(G\) with identity element 1 such that \(G\) has a countable free generating set \(P\) of ``primes'' \(p\) and such that \(G\) admits an integer-valued degree mapping \(\partial\) which satisfies some suitable conditions. Let \(\widetilde{G}(n)\) denote the total number of elements of degree \(n\) in \(G\) and \(\widetilde{P}(n)\) the total number of primes of degree \(n\) in \(G\). The abstract prime number theorem means \(\widetilde{P}(n)= q^n/n+O(q^{\theta n})\) as \(n\to\infty\), where \(0<\theta<1\). Usually the following distribution of the elements of \(G\) is assumed: \(\widetilde{G}(n)= Aq^n+O(q^{\nu n})\) as \(n\to\infty\) with constants \(A>0\), \(q>1\), and \(0\leq\nu<1\), and furthermore, some conditions, for example the generating function \(Z^\#(y)= \sum\widetilde{G}(n)y^n\) of \(\widetilde{G}(n)\) has no zeros on the circle \(|y|=q^{-1}\). By analogy with the theory of Beurling generalized prime numbers it is interesting to investigate the prime element number theorem under more general conditions. In the well written paper under review the author proves some generalizations of the abstract prime number theorem \[ \widetilde{P}(n)\sim \rho_rq^n n^{-1} \] under various conditions on the coefficients in \[ \widetilde{G}(n)= q^n\sum^r_{\nu=1} A_\nu n^{\rho_\nu-1}+ O(q^n n^{-\gamma}). \] The key of his investigations is to determine the number of zeros of the generating function \(Z^\#(y)\) on the circle \(|y|=q^{-1}\), a subject matter which belongs more or less to the theory of holomorphic functions, but it shows a major divergence from the phenomena in the classical zeta function world. Several explicit examples show that in some sense the results are best possible, but present also new arithmetic features.