I have seen the following statement on a Stack Exchange answer:
Let $X = 1 - (1/U - \left\lfloor {1/U} \right\rfloor )$, where $U$ is a uniform random variable in $[0, 1]$. Then— $$\mathbb{E}[X] = \gamma,$$ where $\gamma$ is the Euler–Mascheroni constant.
Is there a reference to a paper that shows this result? I could not find this statement in my searches so far.
The fact that the expectation of $1-\{\frac1U\}$ equals $\gamma$, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part of $x$, is equivalent to the assertion $$ \int_0^1 \bigg\{ \frac1u \bigg\} \,du = 1-\gamma, $$ which can be found on pages 109–111 of Havil's Gamma: Exploring Euler's constant; it can also be found as Corollary 1.15 of Montgomery and Vaughan's Multiplicative Number Theory I. Classical Theory (after the change of variables $u\mapsto \frac1u$). You can also prove it directly with the same change of variables, by writing $\{x\}=x-\lfloor x\rfloor$ and dividing up the resulting integral from $1$ to $\infty$ into infinitely many integrals of length $1$ on which $\lfloor x\rfloor$ is constant.