Nowak, Werner Georg On the average number of finite Abelian groups of a given order. (English) Zbl 0749.11043 Ann. Sci. Math. Qué. 15, No. 2, 193-202 (1991). 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) Cited in 3 Documents MSC: 11N45 Asymptotic results on counting functions for algebraic and topological structures 20K99 Abelian groups Keywords:number of non-isomorphic Abelian groups; sum of reciprocals; multiplicative prime-independent function; asymptotic expansion; error term Citations:Zbl 0442.10032 PDFBibTeX XMLCite \textit{W. G. Nowak}, Ann. Sci. Math. Qué. 15, No. 2, 193--202 (1991; Zbl 0749.11043)