I was reading about the Riemann zeta function in the region $Re(Z) > 1$, where it can be represented by the Euler product formula. And the book mentioned that there can be no zeros in this region, since each factor $1/(1 - 1/p^s)$ is never zero.
Can someone explain why the product can't converge to zero? For example, consider the infinite product $1/2 \cdot 1/3 \cdot 1/4 \cdots$. This seems to converge to zero, but each factor is not zero.
Thanks!
On
For all prime $p$ we have that
$$\frac{1}{1-1/p^s} = \frac{1}{\frac{p^s-1}{p^s}} = \frac{p^s}{p^s-1}>1$$
In that case the terms are not going to $0$.
On
Here they explain your question.
In particular they show
$$\prod_{k=1}^{\infty} \frac{1}{1-\frac{1}{p_k}}=\sum_{n=1}^{\infty} \frac{1}{n^s}>1$$
This argument itself seems indeed wrong, as your counterexample shows.
However, the members in your counterexample tend to $0$, but the members of the Euler product formula tend to $1$ from above (as $p\to\infty$).
'Taking logarithm', the product will become a sum, and in this sum each summand is positive. Probably that was the intended meaning..