What are $z$'s such that $\prod_{n=2}^\infty (1-n^{-z})$ is convergent?
Let $\mathscr{R}$ be the region where the product converges.
Assume that the product is convergent at $z$. Then, $\lim_{n\to\infty} (1-n^{-z})=1$. Thus, $Re(z) >0$. Thus, $\mathscr{R}\subset \{z:Re(z)>0\}$ That's all I can do so far now.. How do I proceed?
It does not seem easy to show the convergence of $\sum Log(1-n^{-z})$.
If $Re(z)>1$, then $\prod (1+ n^{-z})$ is absolutely convergent. Hence $\{z:Re(z)>1\}\subset \mathscr{R}$. However, $\mathscr{R}$ may be strictly larger than the set $\{z:Re(z)>1\}$. Isn't it?