## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  695.10040
Autor:  Erdös, Paul; Ivic, Aleksandar
Title:  On the iterates of the enumerating function of finite Abelian groups. (In English)
Source:  Bull., Cl. Sci. Math. Nat., Sci. Math. 17, 13-22 (1989).
Review:  Let a(n) denote the number of non-isomorphic Abelian groups of order n. It was proved by the reviewer [Q. J. Math., Oxf. II. Ser. 21, 273-275 (1970; Zbl 206.03402)] that

limsupn ––> oo\frac{log a(n) log log n}{log n} = \frac{log 5}{4}.

Now the authors investigate the iterates of a(n), which are defined by a(r)(n) = a(a(r-1)(n)), a(1)(n) = a(n), r = 2,3,... . The main result is

a(2)(n) << \exp{B(log n)7/8/(log log n)19/16}

with a positive constant B and log a(r)(n) << (log n)cr with c1 = 1, c2 = 7/8 and cr \leq (½)cr-1+(3/8)cr-2 for r \geq 3.
Furthermore, let K(n) = max {r: a(r)(n) = 1}. Then an asymptotic representation for the mean value of K(n) is established.
Reviewer:  E.Krätzel
Classif.:  * 11N45 Asymptotic results on counting functions for other structures
Keywords:  arithmetic functions; finite abelian groups; number of non-isomorphic Abelian groups of order n; iterates; asymptotic representation; mean value
Citations:  Zbl 206.034

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