## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  010.29402
Autor:  Erdös, Pál; Szekeres, George
Title:  Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem. (On the number of abelian groups of given order and on a related number-theoretical problem.) (In German)
Source:  Acta Litt. Sci. Szeged 7, 95-102 (1934).
Review:  Let fi(n) denote the number of different (= disregarding the order of the factors) ways in which the integer n can be written as a product, each of whose factors is a prime number raised to a power \geq i. For the partial sums of these fi(n), the authors prove the asymptotic formulae sumk = 1n fi(k) = Ai n1/i+O(n1/i+1); the constant Ai = prodk = 1oo \zeta(1+k/i), where \zeta (s) denotes Riemann's Zeta function. For i = 1, sum f1(k) is the number of finite abelian groups whose orders are \leq n; it is = A1· n+O(n ½). In a second part the authors give similar asymptotic formulae for the frequency of those integers for which fi (n) \ne 0.
Reviewer:  F.Bohnenblust (Princeton, N.J.)
Classif.:  * 11N45 Asymptotic results on counting functions for other structures
20K01 Finite abelian groups
Index Words:  number theory

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