Convergence of product over all primes

Question

How can we find the values of $x$ for which $$\prod_{p \text{ prime}}{1-\frac{x^2}{p^2}}$$ converges? I know that this product $$\prod_{p \text{ prime}}{1+\frac{x^2}{p^2}}$$ converges if and only if $$\sum_{p \text{ prime}}{\frac{x^2}{p^2}}$$ converges, and since $$\sum_{p \text{ prime}}{\frac{x^2}{p^2}}=\sum_{p \text{ prime}}{x^2\frac{1}{p^2}}=x^2\sum_{p \text{ prime}}{\frac{1}{p^2}}$$ we can see that this sum is a number $x^2$ times primezeta(2), which converges, so the product converges for all $x$. Is this correct? Is there a way to adapt this to the original product? Any help would be appriciated!

Answer 1

This should converge for all $x$; apart from finitely many terms at the outset all terms are in $(0,1)$ so the sequence from that point on is monotonic and bounded by zero. The partial products are a monotonic bounded sequence of real numbers, which must therefore converge.