Physics and the Apéry constant, an example for mathematicians The Wikipedia's entry for the Apéry's constant tell us that the Apéry's  constant $$\zeta(3)=\sum_{n=1}^\infty\frac{1}{n^3}$$ arises in physical problems.

Question. Can you tell us, from a divulgative viewpoint but with mathematical details if it is possible, a nice physical problem involving the Apéry's  constant? Many thanks. 

I believe that it is a curiosity, but and you know mathematics but and an example of such concise problem in physics (see the problems, or other, that refers Wikipedia if you know to show/explain us the calculations after your introduction to the physic problem), then your answer should be nice for all us.

Zeta values appear in a large number of physical applications. Just to point out a relevant field, one of these is the study of black bodies, an issue that can be extended to the larger theory of fundamental particles such as photons, electrons and positrons. For example, a  field where both $\zeta (3) $ and $\zeta (4) $ are commonly used is to quantify black body energy radiation. A black body of surface $S $ and temperature $T$ radiates energy at a rate equal to $\sigma ST^4$, where 
$$\sigma =\frac {2 \pi^5}{15 } \frac{k^4}{h^3c^2} = 12 \pi \zeta (4) \frac{k^4}{h^3c^2}=5.67 \cdot 10^{-8} \text {J }\, \text { m}^{-2}\,\text { s}^{-1} \,\text { K}^{-4}      $$
is the Stefan-Boltzmann constant, defined by the Planck's constant $h $, the speed of light $c $, and  the Boltzmann's constant $k $. The presence of $\zeta (4) $ results from the integral 
$$\int_0^\infty \frac {2 \pi x^3 dx}{e^x-1}=12 \pi \zeta (4) \approx 40.8026...$$
calculated over the black body spectrum (here is the numerical estimation of the integral by WA for $x=0$ to $10^6$). A similar expression, given by $\sigma' ST^3$, provides  the rate of emission of photons over time by a black body. In this case, $\sigma'$ is given by  
$$\sigma'=4 \pi \zeta (3) \frac{k^3}{h^3c^2}   $$
where the presence of the Apéry's constant $\zeta (3) $  results from the integral 
$$\int_0^\infty \frac {2 \pi x^2 dx}{e^x-1}=4 \pi \zeta (3)   \approx 15.1055...$$
again calculated over the black body spectrum (here is the numerical estimation by WA). 
As a confirmation of the extension of these concepts to the study of subatomic particles, another similar expression including the Apéry's constant gives the estimated average density of photons for the cosmic microwave background radiation, given by 
$$16 \pi \zeta (3) \left ( \frac{kT_0}{hc} \right)^3 \approx 413 \, \text {cm}^{-3}   $$
where $T_0$ is the temperature of the radiation. A nice derivation of this result is provided here.
