I understand how to derive the Euler Product over primes for the Zeta function using the sieving method. However, upon reading more it said the following..
$$\prod_{P Prime}^{\infty}(1-p^{-s})^{-1} = \prod_{P Prime}^{\infty}(\sum_{n=1}^{\infty}p^{-ns})= \sum_{n=1}^{\infty}\frac{1}{n^s} $$
So my question is , how is it possible to directly prove that the product of that geometric series is equal to the zeta function sum? Does the use of the geometric series involve a different proof than the one usually given which uses the sieving?
Thank you for your help a lot.