I have 2 questions"
It's often said the appropriate sum converges as long as $s$ has real part more than $1$. Are we deciding on the branch of $k=0$ here: $n^s=e^{(ln(n)+2\pi ik)s}$? If not, why does the sum converge?
Why is the dirichlet series analytic inside of where it converges (again I'm assuming we fix $k=0$?