The prime element theorem on additive formations (Q1298010)

Since A. Beurling 1937 published his note on the prime number theorem for generalized primes there were also many papers in which generalizations of the prime number theorem for arithmetic progressions were considered. So, in 1954 \textit{W. Forman} and \textit{H. N. Shapiro} [Commun. Pure Appl. Math. 7, 587-619 (1954; Zbl 0057.28404)] studied the multiplicative case, where \textit{F. Halter-Koch} and \textit{R. Warlimont} in 1991 developed the concept of additive formations based on additive arithmetic semigroups, introduced by \textit{J. Knopfmacher}. In the note under review the author gives answers to the problem of which condition on the distribution of all elements of an additive formation in congruence classes guarantees the nonvanishing of the associated zeta function \(Z(z,\chi)\) on the circle \(|z|=q^{-1}\). Similar to the classical situation this is the key to establish a prime element theorem of the form \[ \pi_\alpha(m) = c \frac{q^m}{m}+ O\left(\frac{\xi^m}{m}\right), \] where \(\pi_\alpha(m)\) counts the number of prime elements \(p\) of degree \(\partial(p)=m\) in the congruence class \(\alpha\), \(q > \xi \geq q^{\frac 12}\). It is remarkable, that the above constant \(c\) is independent of \(\alpha\). (We drop here the exact definition of an additive formation and the assumed Axiom B, which stated that \[ \gamma_\alpha(m)=\sum_{a\in\alpha, \partial(a)=m}1=Aq^m+O(s^m) \] with some constants \(A>0\), \(q>1\) and \(s\) with \(0\leq s<q\)). The condition under which no zeta function \(Z(z,\chi)\) vanishes on the closed disk \(|z|\leq q^{-1}\) is of the form \[ \sum_\alpha\sum_m\left(\gamma_\alpha(m)-cAq^m\right)^2 q^{-m}<\infty, \] a theorem which is sharp in some sense. Several examples especially from algebraic function fields of transcendance degree one illustrate the validity of his results.