A simple proof of Euler's product formula
Introduction:
The Euler product formula states that if is the Riemann zeta function and is prime:
\begin{equation} \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p} \frac{1}{1-p^{-s}} \end{equation}
holds for all such that is absolutely convergent.
Proof:
Every positive integer has a unique prime factorization:
\begin{equation} \forall n \in \mathbb{N^*} \exists c_p \in \mathbb{N}, n = \prod_p p^{c_p} \end{equation}
where .
Furthermore, we note that:
\begin{equation} \prod_p \frac{1}{1-p^{-s}} = \prod_p \big(\sum_{c_p = 0}^\infty p^{-c_p s} \big) \end{equation}
due to elementary properties of geometric series.
In the formal expansion (2) we note that each term has a unique prime factorization and that every possible prime factorization occurs once. It follows that if converges absolutely we may rearrange the sum however we wish and so:
\begin{equation} \zeta(s) = \prod_{p} \frac{1}{1-p^{-s}} \end{equation}
provided that the hypotheses on are satisfied.