Divergence of $\zeta(z)$ tamed or not tamed by any analytic continuation

Question

We know the conjecture about the Riemann hypothesis is about the nontrivial zeros are on $$(1/2 + r i)$$ for some $r \in \mathbb{R}$ of Riemann zeta function.

My question is how many divergences of $\zeta(z)$ can be tamed by analytic continuations and how many divergences of $\zeta(z)$ canNOT be tamed by analytic continuations?

For example, $$\zeta(1)=1+2+3+4+\dots=\infty$$ seem not to be tamed by any analytic continuation. How about other $\zeta(z)$? Are there others cannot be done by analytic continuation as a finite value?

Answer 1

The only pole of the Riemann zeta function is at $ z = 1 $, so that's the only divergence that's not "tamed" by analytic continuation.

Answer 2

The series $\sum_{n\ge 1}n^{-s}$ diverges whenever $\Re(s)\le 1$,

Iff $s=1$ or $\Re(s)<1$ then $\lim_{N\to \infty}|\sum_{n\ge N}n^{-s}|= \infty$.

For $\Re(s)\le 1$,$s\not\in \Bbb{R}$ then $\frac{\sum_{n\ge N}n^{-s}}{|\sum_{n\ge N}n^{-s}|}$ oscillates.

For $\Re(s)\le 1,s\ne 1$ $$\lim_{N\to \infty} \frac{\sum_{n\ge N}n^{-s}}{\int_1^N x^{-s}dx}=\lim_{N\to \infty} \frac{1-s}{N^{1-s}}\sum_{n\ge N}n^{-s}=1$$

It is mostly similar for $\eta(s)$ with $\Re(s)=0$ instead of $\Re(s)=1$.

For the analytic continuation: you really have to compare with $\frac1{1-s}$ which is the analytic continuation of $\sum_{n\ge 0} s^n,|s|<1$ to $\Bbb{C}-\{1\}$, the setting is exactly the same for $\zeta(s)$.

Next, consider the functional equation saying that $\pi^{-(1/2+s)/2}\Gamma((1/2+s)/2)\zeta(1/2+s)$ is an even function which is real-valued for $s$ on the imaginary axis, thus having a zero at each sign change (the non-trivial zeros, under the RH).