# Analytic continuation vs. series convergence near convergence boundary

Citing Wikipedia, the Riemann zeta function is the analytic continuation of

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$

The series itself is only convergent in the right half complex plane where $Re(s)>1$.

What I would like to understand is how the series behaves near the line $1+it$, i.e. the boundary of this right half plane. On the boundary line we have $|\zeta(1+it)|<\infty$. Lets look at the specific example of $t=5$. for real $\epsilon>0$ I would write

$$\zeta(1+5i) = \lim_{\epsilon\to 0}\sum_{n=1}^{\infty} \frac{1}{n^{1+\epsilon+5i}}$$

and I assume that in this order of taking the two limits, this is actually a correct equation, since otherwise $\zeta$ would not be an analytic continuation.

Does this mean that $\sum_{n=1}^{\infty} \frac{1}{n^{1+\epsilon+5i}}$ is bounded for each $\epsilon>0$? But if it is bounded yet does not converge for $\epsilon\to0$, does it mean that the sum, if understood as $\lim_{n\to\infty} \sum^n$, oscillates around the value of $\zeta(1+5i)$? What happens when $\epsilon$ goes towards $0$?