The prime element theorem in additive arithmetic semigroups. II
The author continues his investigations on prime element theorems in additive semigroups which he started in part I [Ill. J. Math. 40, 245-280 (1996; Zbl 0863.11065)]. The main purpose is to prove stronger estimates of the remainder terms, i.e. instead of \(o(1)\)-type now more explicitly some \(O(1)\)-type. An additive arithmetic semigroup is understood as 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 \(G\), \(\widetilde P(n)\) the total number of primes of degree \(n\) in \(G\), and \(\widetilde\Lambda(n)=\sum_{r| n}r\widetilde P(r)\). To give a typical example (Theorem 1.4) of his results, assuming \[ \widetilde G(n)= A q^n+O(q^{n}n^{-\gamma})\quad \text{ as}\quad n\to\infty \] with \(A>0\), \(q>1\), and \(\gamma>2\) the author shows that either \[ \widetilde\Lambda(n)= q^n\bigl(1+O(n^{2-\gamma})\bigr)\quad\text{ or}\quad q^n\bigl(1+O(n^{2-\gamma}\log n)\bigr), \] according to whether \(\gamma\) is not or is an integer, or \[ \widetilde\Lambda(n)= q^n\bigl(1-(-1)^n+O(n^{2-\gamma}\log n)\bigr)\quad\text{ or}\quad q^n\bigl(1-(-1)^n+O(n^{2-\gamma}\log^2 n)\bigr). \] Which of these possibilities are valid depends on the fact whether the generating function \(Z^{\#}(y)=\sum_n\widetilde G(n) y^n\) has a zero on the circle \(| y| =q^{-1}\) or not. The major part of this interesting paper deals with the delicate analysis of the behavior of \(Z^{\#}(y)\) on the boundary \(| y| =q^{-1}\), where some kind of Lipschitz conditions occur. An example illustrates that it may happen (in contrast to the classical situation) that the associated \(\mu\)-function does not have a mean value because of the dominant perturbation of the zero at \(y=-q^{-1}\).
scientific article; zbMATH DE number 1191158