On the zeta-function of an arithmetical semigroup (Q1434499)

The concept of a commutative additive arithmetical semigroup \(A\) was introduced by John Knopfmacher. Associated to \(A\) there is a Zeta function \(Z(x)\) which reflects the arithmetical properties of \(A\). It is defined by \[ Z(x)=\sum_{n=0}^\infty f(n)x^n=\prod_{n=1}^\infty(1-x^n)^{-g(n)}, \] where \(f(n)\) resp. \(g(n)\) counts in some sense the elements resp. the generators of \(A\). It is known, that \(g(n)=o(f(n))\) implies that the radius of convergence \(\rho\) of \(Z(x)\) is positive and that \(Z(\rho)\) diverges. On the other hand there are also multiplicative arithmetical semigroups with analogous Zeta functions, but now defined as Dirichlet series, and with a similar result on the abscissa of convergence. In the paper under review the authors gives unified proofs of both results which rest on his theorem about power series. The author points out that his theorem is indeed a corollary of a theorem of W. Rudin.