Show that the series $\sum_{n\geq 1}n^{-z}$ converges (locally) normally on the halfplane

Question

Show that the series $\sum_{n\geq 1}n^{-z}$ converges (locally) normally on the halfplane $({z \in\mathbb{C}: Re(z)>1})$

I am stuck with this part question. I just do not properly know how to show local normal convergence in this example. Any help is greatly appreciated.

Answer 1

Take $a\in(1,+\infty)$. Then every compact subset $K$ of your halfplane is a subset of $\left\{z\in\mathbb{C}\,\middle|\,\operatorname{Re}z\geqslant a\right\}$, for some such $a$. But the series $\sum_{n=1}^\infty n^{-a}$ converges and$$(\forall z\in K):|n^{-z}|=n^{\operatorname{Re}(-z)}\leqslant n^{-a}.$$Therefore, by the Weierstrass $M$-test, your series converges uniformly on $K$.