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On the average number of finite Abelian groups of a given order. (English) Zbl 0749.11043

Let \(a(n)\) denote the number of non-isomorphic Abelian groups of order \(n\), J.-M. De Koninck and A. Ivić [Topics in arithmetical functions (North Holland 1980; Zbl 0442.10032)] established an asymptotic formula for the sum \(\sum_{n\leq x} 1/a(n)\). In this article, a more precise asymptotic expansion with an error term which is best possible on the basis of the present knowledge about the zeros of the Riemann zeta- function is established. This result is obtained as a special case of a more general theorem which applies to all multiplicative and prime- independent arithmetic functions \(a(n)\) which satisfy \(a(p)=1\) for each prime \(p\) and \(a(n)\geq 1\) for every positive integer \(n\).
Reviewer: Lu Minggao (Hefei)

MSC:

11N45 Asymptotic results on counting functions for algebraic and topological structures
20K99 Abelian groups

Citations:

Zbl 0442.10032
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