On the number of abelian groups of given order and on a related number-theoretic problem. (Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem.) (German) Zbl 0010.29402

Let \(f_i(n)\) 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 \(f_i(n)\), the authors prove the asymptotic formulae \[ \sum_{k=1}^n f_i(k) = A_i n^{1/i}+O(n^{1/i+1}); \] the constant \(A_i = \prod_{k=1}^{\infty} \zeta(1+k/i)\), where \(\zeta (s)\) denotes Riemann’s zeta-function. For \(i=1\), \(\sum f_1(k)\) is the number of finite abelian groups whose orders are \(\leq n\); it is \(= A_1\cdot n+O(n^{1/2})\). In a second part the authors give similar asymptotic formulae for the frequency of those integers for which \(f_i (n) \neq 0\).


11N45 Asymptotic results on counting functions for algebraic and topological structures
20K01 Finite abelian groups