Calculation of $\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\left\{\frac{y}{x}\right\}\left\{\frac{z}{y}\right\}\left\{ \frac{x}{z}\right\}\,dx\,dy\,dz$

Question

I 'd like to calculate the integral of the following expression in cube $$\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\left\{\frac{y}{x}\right\}\left\{\frac{z}{y}\right\}\left\{ \frac{x}{z}\right\}\,dx\,dy\,dz$$

When I calculate this integral I finally find that this integral has the relation with the infinite sum of $1/n^3$ But I don't know the exact way to calculate this some. Any one can help me to find the answer?