On the distribution of the degrees of prime element factors in additive arithmetical semigroups (Q1969705)

By an additive arithmetic semigroup \((G,\partial)\) it is understood a free commutative semigroup \(G\) with identity element \(e\) 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 \(\gamma(n):=\#\{a\in G\mid \partial(a)=n\}\) and \(\pi(m):=\#\{p\in P\mid \partial(p)=m\}\). The standard example is the multiplicative semigroup of all monic polynomials \(f(X)\in F_q[X]\) with \(\partial(f(X))=\text{degree} f(X)\). While \textit{A. Knopfmacher} [Discrete Math. 196, 197-206 (1999; Zbl 0931.11055)] considered the behaviour of \[ \rho(f(X)):=\#\{m\in{\mathbb N}\mid \prod_{p^{\varepsilon_p(f)}\parallel f, \partial(p)=m} p^{\varepsilon_p(f)} \not=e\}, \] in the paper under review the analogous problem is studied in a more general setting, namely for arithmetical semigroups \((G,\partial)\), which satisfy the so called Axiom A: There are constants \(A>0\), \(q>1\) such that \[ \sum_{n=0}^\infty \sup_{m\geq n}|\gamma(m)q^{-m}-A|<\infty. \] It is shown among other results that \[ \gamma(n)^{-1}\sum_{\partial(a)=n}\rho(a) = \sum_{m=1}^n\tfrac 1m + c_1+O(t(n)\log n +\tfrac 1n) \] with some explicitly given quantities \(c_1\), \(t(n)\). Axiom A was recently introduced by W. B. Zhang and is the weakest condition to impose on \((G,\partial)\) such that a Chebyshev type upper estimation for \(\pi(m)\) holds true. The proofs are elementary but need a very careful analysis of the involved series and some (not yet published) results of W. B. Zhang.