Some remarks on additive arithmetical semigroups (Q1873246)

Under weak conditions called Axioms \({\mathcal{A}}_1\) and \({\mathcal{A}}_2\) the author proves Chebyshev-type estimates and asymptotic formulas around the prime number theorem, respectively, in general additive arithmetical semigroups. The new Axioms \({\mathcal{A}}_1\) and \({\mathcal{A}}_2\) depend on the analytic behavior of the associated zeta-function in the disc \(|z|<q^{-1}\). As applications he derives asymptotic laws for prime divisor functions and the distribution of prime factors of distinct degrees.