Double Integrals · Superchallenge problem

For real $u$ let $\{u\} = u - \lfloor u \rfloor$ be its fractional part, and let $\gamma = \displaystyle\lim_{n\to\infty}\Big(1 + \tfrac12 + \cdots + \tfrac1n - \ln n\Big)$ be the Euler–Mascheroni constant. Evaluate $\displaystyle\int_0^{1}\!\int_0^{1} \left\{\dfrac{x}{y}\right\}dx\,dy$ and give its value as a decimal rounded to $4$ decimal places (use $\gamma \approx 0.5772157$). The exact value can be written in terms of $\gamma$.