Über ein Problem mit Fareybrüchen. (German) Zbl 0081.27204
Österr. Akad. Wiss., Math.-Naturw. Kl., Anz. 1957, 267-268 (1957); Nachtrag 1958, 222-223 (1958).
Let \(I_i\) denote for \(i = 1, \ldots, n\) the interval \(((i - 1)/n, i/n)\). Let \(p_i/q_i\) be the fraction of the Farey series of order \(n\), with minimum denominator, which lies in \(I_i\). It is proved without using estimates for arithmetic functions, that
\[ n^{-1} \sum_{i=1}^n q_i^{-1} \rightarrow 0\quad\text{as }n\to\infty.\]
\[ n^{-1} \sum_{i=1}^n q_i^{-1} \rightarrow 0\quad\text{as }n\to\infty.\]
Reviewer: W. Verdenius
MSC:
| 11B57 | Farey sequences; the sequences \(1^k, 2^k, \dots\) |
