Limit of the product is infinity

Question

How can we proof it? $\displaystyle\lim_{n\to\infty}\prod_{p_i\leqslant{n}}\frac{1}{1-\frac{1}{p_i}} = \infty$ Thanks

Answer 1

This is a consequence of the Euler product formula and the fact that $\sum_{n=1}^\infty \frac{1}{n}$ diverges.

Answer 2

If $p$ is prime then that product is the Euler product formula for the Riemann zeta function.

$\displaystyle \prod_{p \hspace{1 mm}prime} \frac{1}{1-p^{-s}}=\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}=\zeta(s)$. For $s=1$ you get the harmonic series which is divergent so your product is divergent.