It is said that the Euler product $\prod_p \frac{1}{1-p^{-s}}$ diverges as $s \to 1^+$ proves we can't find constants $C$,$\theta$ with $\theta < 1$ such that $\pi(x) < C x^\theta$ because that would imply that the product converges for $s > \theta$.
I don't understand this deduction at all, how is the fact about the prime counting function concluded?