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Some applications of Franel-Kluyver’s integral. II. (English) Zbl 0671.10002
[Part I, cf. Acta Math. Univ. Comenianae 50/51, 237–246 (1987; Zbl 0667.10023).]
The author uses methods from uniform distribution and an integral identity proved by Kluyver to give the following representation for the greatest common divisor \((a,b)\) of two arbitrary positive integers \(a\) and \(b\):
\[ \frac{1}{12}\cdot \frac{(a,b)^ 2}{ab}=\sum^{\infty}_{k=2}\frac{1}{2(2k-1\quad)}\cdot \frac{(2k)!}{(2^ k k!)^ 2}\cdot \sum_{r,s=1,...,k;2\leq r+s\leq k}\frac{1}\quad {x^{2(r+s)-2}}\cdot \]
\[ \cdot \left( \begin{matrix} 2(r+s)\\ 2r\end{matrix} \right)\cdot \frac{B_{2r}}{a^{2r-1}}\cdot \frac{B_{2s}}\quad {b^{2s-1}}\cdot \frac{-2}{2(r+s)(2(r+s)-1)}\cdot [(-1)^{r+s-1} \left( \begin{matrix} \quad k\\ r+s-1\end{matrix} \right)- \]
\[ (-1)^ k\cdot 2^{2k-2(r+s)+2} \left( \begin{matrix} k\\ 2k-2(r+s)+2\end{matrix} \right)] \] where \(x\) is an arbitrary positive integer and \(B_ r\) denotes the \(r\)th Bernoulli number. Further a result on the speed of convergence of the infinite series is given.
Reviewer: G.Larcher

MSC:
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11J71 Distribution modulo one
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References:
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