What is the value of this triple integral $$\int\limits\_0^{2\pi}\int\limits\_0^{2\pi}\int\limits\_0^{2\pi}|\cos x+\cos y+\cos z|\ dx\ dy\ dz?$$ It has to do with some Schwarz lemma.



An interesting triple integral
**Comment**  

This is related to some papers (e.g. [this](https://carma.newcastle.edu.au/jon/walks2.pdf)) by P.M. Borwein et. al. on short random walks in the plane.


Let $X\_1, X\_2, \dots$ be i.i.d. random variables, uniformly distributed on the unit circle $|z|=1$ in the complex plane. Then
$$
X\_1+X\_2+X\_3
$$
is a random variable in the plane, and your integral is
$$
(2\pi)^3\;\mathbb{E}\Big[\big|\mathrm{Re}(X\_1+X\_2+X\_3)\big|\Big]
$$


Borwein and collaborators have information on moments
$$
W\_3(s) := \mathbb{E}\Big[\big|X\_1+X\_2+X\_3\big|^s\Big]
$$
including "closed form" in terms of hypergeometric functions when $s \in \mathbb N$. In particular
$$
W\_3(1) = \frac{3}{16}\;\frac{2^{1/3}}{\pi^4}\Gamma({\textstyle \frac{1}{3}})^6
+
\frac{27}{4}\;\frac{2^{2/3}}{\pi^4}\Gamma({\textstyle \frac{2}{3}})^6 .
$$


Now, if the distribution of
$$
Y = X\_1+X\_2+X\_3
$$
is rotationally symmetric in the complex plane, and we have the exact value of $\mathbb E[|Y|]$, can we find $\mathbb E[|\mathrm{Re}\;Y|]$ ??


