# Proving the golden key,

1. Apr 24, 2006

### heartless

Hello,
I'm trying to prove that

$$\sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p} (1-p^ {-s} )^ {-1}$$

I know why it is and a proof, but I'm actually looking for
a different way to prove going backward and deriving the
sum from the product of primes. Can you show me a way to do that?
$$\prod_{p} (1-p^ {-s} )^ {-1} = ...$$

Thanks,

p.s that -s above and then following -1 should be both exponents for the equation
same with -s and -1 on the bottom, I'm not the master latex writer. Can somebody also tell me why it doesn't work?

Last edited: Apr 24, 2006
2. Apr 25, 2006

### SirArthur333

The golden key lives in the golden house with goldielocks. It is guarded by two dragons. Penitent man will pass.

3. Apr 25, 2006

### heartless

I will never pass... but! I wouldn't have to, if you show me the very proof :)

4. Apr 25, 2006

### shmoe

What proof do you know? The usual is to look at the finite product $$\prod_{p\leq x}(1-p^{-s})^{-1}$$ and expand using geometric series. Then compare with $$\sum_{n\leq x} n^{-s}$$, and show the difference the two goes to zero as x-> infinity.