Prime Numbers and pi

[Mart]

I came across a formula by Euler which was a derivation of (IIRC) pi^2 which used an infinite series consisting of the prime numbers. Does anyone know about this or can anyone point me to a useful site?

[shmoe]

Do you mean:

 

[math]\frac{\pi^2}{6}=\sum_{n=1}^\infty n^{-2}=\prod_{p\ \text{prime}}(1-p^{-2})^{-1}[/math]

 

Euler argued the first equality by comparing coefficients of the taylor series of sin(x) with it's product form (which hadn't been fully justified until Hadamard). Here's a bunch of different ways to prove it (Euler's is #7):

 

http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf

 

The second equality above is just a special case of the Euler product form for the Riemann Zeta function (<-words to punch into google). It requires unique factorization and some arguments about convergence.

[Mart]

Thanks shmoe.