The Cauchy principal value of the Riemann Zeta function

Question

In many online sources, you can find
$$ \zeta(s) \overset{C.P.}{=} \lim_{\epsilon \to 0} \left(\frac{\zeta(s+\epsilon)+\zeta(s-\epsilon)}{2}\right).
$$
This seems quite logical, but I neither know how to derive it nor how to derive the second equality
$$
\zeta(1) \overset{C.P.}{=}\lim_{\epsilon \to 0} \left(\frac{\zeta(1+\epsilon)+\zeta(1-\epsilon)}{2}\right)=\gamma
$$
(where $\gamma$ is the Euler-Mascheroni-constant).
I always knew the Cauchy principal value as a a method for assigning values to improper integrals, and although I know many integral representations of the Riemann Zeta function, I dont't really know how to show this specific result. Many thanks in advance!

Answer 1

The asymptotic

$$ \zeta(s) = \frac{1}{s-1} + \gamma + O(|s - 1|) $$

for the Riemann zeta function near $ s = 1 $ is well known, and it immediately implies your result, since the $ 1/(s-1) $ terms cancel when computing the limit in your principal value.

For a proof of this asymptotic, see this post.