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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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