Mart Posted October 13, 2005 Share Posted October 13, 2005 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? Link to comment Share on other sites More sharing options...

shmoe Posted October 14, 2005 Share Posted October 14, 2005 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. Link to comment Share on other sites More sharing options...

Mart Posted October 14, 2005 Author Share Posted October 14, 2005 Thanks shmoe. Link to comment Share on other sites More sharing options...

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