The evaluation with Maple suggests the triple integral is around $1$, but Mathematica tells
it's $0.0958758$. When using the code
N[Integrate[FractionalPart[x/y] FractionalPart[y/z] FractionalPart[z/x], {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]]
it returns:
"NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>
NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 0.09587582633950331and 0.004696948831748114
for the integral and error estimates. >>"
I'd like to know if Mathematica's approximation is correct, and if there is a possible closed form.