Analytic number theory in formations based on additive arithmetical semigroups

The prime element theorem on additive formations

“…In particular, in [6], he introduced the concept of additive arithmetic semigroups and proposed the problem to develop a theory of formations and L-functions based on additive arithmetic semigroups (for a complete survey, refer to [3]). …”

“…This problem becomes the main subject of the paper [3] by F. HalterKoch and R. Warlimont. In particular, they define the concept of additive formations based on additive arithmetic semigroups.…”

“…Then, let Z(z, χ) := a∈G χ(a)z ∂(a) , (1.2) where the summation is taken over all a ∈ G in an ascending order of ∂(a). As in [3], Z(z, χ) is called a zeta function on the formation (G, ∂, ∼). For an additive formation (G, ∂, ∼) satisfying Axiom B, Z(z, χ) is holomorphic and has no zeros in the open disk {|z| < q −1 } due to an analogue of the Euler product formula.…”

“…Then, Theorem 4.1, a prime element theorem, follows. We notice that the statement of Proposition 7 in [3] contains incorrect formulas. Hence, we shall give a correction of Proposition 7 and, besides, a direct proof which leads to a remainder of the weaker form (1.3), without appealing to A. Weil's theorem.…”

“…Hence, we shall give a correction of Proposition 7 and, besides, a direct proof which leads to a remainder of the weaker form (1.3), without appealing to A. Weil's theorem. To show this, we shall have to sketch a proof of Lemma 4.4, which is essentially Lemma 4 in [3], because the latter has not been stated and proved correctly there.…”