Can anyone offer some clarification on the following equivalence? Perhaps as happens too often, I was told the statement follows from 'basic analysis', and while this is probably true, I can't seem to get it to work! Here is the claim:
For all $s\in \mathbb{C}$ with Re($s$)$>1$, the product $ \displaystyle\prod_{p \text{ prime}} \frac{1}{1-p^{-s}} $ converges if and only if the sum $\displaystyle\sum_{p \text{ prime}} {p^{-s}}$ converges.