## 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 $$f_i (n) \neq 0$$.

### MSC:

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

### Keywords:

number of abelian groups of given order