## 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

Zbl 0442.10032
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