TITLE: Physics and the Apéry constant, an example for mathematicians
QUESTION [8 upvotes]: 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.

REPLY [3 votes]: The $\zeta(3)$ constant appears in fluid mechanics as the added mass of a sphere approaching a wall, such as a raindrop (Weihs & Small, 1975), in the form $3\zeta(3) -2$. 
Weihs, D.; Small, R. D., An exact solution of the motion of two adjacent spheres in axisymmetric potential flow, Israel J. Technol. 13, 1-6 (1975). ZBL0318.76010.