$f_k(n)$ denote the number of representation of $n$ as a product of $k$ factors greater than unity. Let $f(1)= 1$, $f(n)$ be the number of representaitons of $n$ as a product of factors greater than unity. (Order is essential).
We have $\sum_{n=2}^\infty \frac{f_k(n)}{n^s} = (\zeta(s)-1)^k$ for $\sigma > 1$. Hence, $$ \sum_{n=1}^{\infty} \frac{f(n)}{n^s} = 1 + \sum_{k=1}^\infty (\zeta(s)-1)^k = 1 + \frac{ \zeta(s)-1}{ 1 - (\zeta(s)-1)} =\frac{1}{2 - \zeta(s) }$$ $\zeta(s)=2$ for $s= \alpha$, $\alpha > 1$. So $|\zeta(s)| < 2 $ for $\sigma > \alpha$, so that equation holds for $\sigma > \alpha$. (p7, Titchmarsh)
What I don't understand is we require $|\zeta(s)-1| < 1$ for geometric series to be written in closed form - and we are given $|\zeta(s)|<2$. So why does the itacilized line hold?