TITLE: Can the product of infinitely many elements from $\mathbb Q$ be irrational?
QUESTION [94 upvotes]: I know there are infinite sums of rational values, which are irrational (for example the Basel Problem). But I was wondering, whether the product of infinitely many rational numbers can be irrational. 
Thank you for your answers.

REPLY [8 votes]: Consider the Riemann-Zeta Function:
$$
\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}.
$$
For $s=2$, the infinite sum on the left is $\pi^{2}/6$, which is irrational. Thus, $\pi^{2}/6$ is an infinite product of rationals.