More specifically, let $(a_n)_n$ be a bounded sequence of complex numbers. Show that for each $\epsilon>0$, the series $\sum\limits_{n=0}^{\infty} a_{n}n^{-z}$ converges uniformly for $\Re z \geq 1+\epsilon$, choosing the principal branch of $n^{-z}$.
It seems like we can bound the series by $B \sum\limits_{n=0}^{\infty} n^{-z}$, but how do we deal with the complex square root?