\[\int_0^1 \int_0^1 \int_0^1 \left(\left\{\frac{x}{y}\right\}\left\{\frac{y}{z}\right\}\left\{\frac{z}{x}\right\}\right)^2 \; dx\; dy\; dz\]

The integral above is equal to \[A-\dfrac{\pi^2}{B}-\dfrac{\zeta(C)}{D}+\dfrac{\pi^2\zeta(E)}{F}+\dfrac{\pi^4 \zeta(G)}{H}+\dfrac{\pi^6}{I}+\dfrac{\zeta^2 (J)}{K}\] for positive integers \(A, B, C, D, E, F, G, H, I, J, K.\)

What is \(A+B+C+D+E+F+G+H+I+J+K?\)

\(\)

**Notation:** \(\{ \cdot\}\) denotes the fractional part function.

**Note:** The solution of this particular problem isn't available anywhere on the web. This is termed as an open problem in the book Fractional Parts, Series, Integrals by Springer.

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