TITLE: What are the connections between pi and prime numbers?
QUESTION [34 upvotes]: I watched a video that said the probability for Gaussian integers to be relatively prime is an expression in $\pi$, and I also know about $\zeta(2) = \pi^2/6$ but I am wondering what are more connections between $\pi$ and prime numbers?

REPLY [26 votes]: Leonard Euler discovered many years ago that
$\frac π 4 = \frac 3 4 \cdot \frac 5 4 \cdot \frac 7 8 \cdot \frac {11} {12} \cdot \frac {13} {12} \cdots$ 
where the numerators on the right-hand side are the odd prime numbers and each denominator (on both sides) is the multiple of 4 nearest to the corresponding numerator. Pretty fascinating if you ask me.