TITLE: Where does $\pi^2$ appear spontaneously within Physical Phenomenon and Mathematics Equations?
QUESTION [22 upvotes]: The term $\pi$ is found to appear in many equations and natural phenomenon; however my question is related to $\pi^2$. 
While trying to figure out the reason for some $\pi^2$ terms appearing in certain equalities that I came across, I have a question. And the question is this: 

In which all mathematics/physics equation or contexts does $\pi^2$ appear inherently? 

-- and (now this second part is merely a follow up question that did not form part of the original query but added later)  where that $\pi^2$ term can lend some interpretation of the underlying phenomenon, just like does $\pi$ whereby we can interpret (in most cases i.e.) that some type of circular ambulation in 1 dimension is involved?? 
As you can understand, the $\pi^2$ term is more complex and does not directly lend itself to an interpretation -- as opposed to $\pi$ which is very intuitive. 
Thanks

REPLY [6 votes]: The $\text{Riemann Zeta function} \ \  \zeta(s)=\large \sum_{n=1}^{\infty} \frac{1}{n^s}$ is used in many branch of Science and mathematics. Replacing $s=2$, we have $ \zeta(2)=\sum \frac{1}{n^2}$. In $1735$ Leonard Euler showed that $$ \zeta(2)=\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{{\color{blue}{\pi^2}}}{6}.$$
This result leads to Number Theory and probability result as follows:
The $ \ {\color{blue}{ probability}} \ $ of two random number being $ \ {\color{blue}{ relatively \ \ prime}} \ $ is given by the following product over all primes
$$ \prod_{p}^{\infty} \left(1-\frac{1}{p^2} \right)=\left(\prod_{p}^{\infty}\frac{1}{1-p^{-2}} \right)^{-1}=\frac{1}{1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots}=\frac{1}{\zeta(2)}=\frac{6}{{\color{blue}{\pi^2}}} \approx 61 \%.$$ 
This is an interesting result where $\pi^2$ is involved.