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On the average number of direct factors of a finite abelian group. (English) Zbl 0113.25305

For positive integers \(n\), let \(\tau(n)\) denote the number of divisors of \(n\); let \(t(n)\) denote the number of decompositions of \(n\) into two relatively prime factors. Dirichlet and Mertens, respectively, obtained the classical estimates for the summatory functions of these functions. In this paper the author derives analogues of these results for finite abelian groups where multiplication in the semigroup of positive integers is replaced by the direct product in the semigroup of finite abelian groups. An analogous approach is used to derive elementary estimates with the Erdős-Szekeres estimate for the number of abelian groups of order \(\leq n\) an important tool. A more penetrating treatment of the summatory functions then yields improved error terms.
Reviewer: W. E. Briggs

MSC:

11N45 Asymptotic results on counting functions for algebraic and topological structures
11N37 Asymptotic results on arithmetic functions
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