This question pertains to the following quote from chapter 7 section 9 of "Summing it Up: From One Plus One to Modern Number Theory" by Avner Ash and Robert Gross:
"What makes the Riemann hypothesis so hard to prove or disprove is the mysterious nature of analytic continuation. The function $\zeta(s)$ is completely determined for all $s\ne 0$ by its values in a small open disc about $s=2$ (for example), but how exactly this determination works is opaque."
For a small circular open disc centered around $s=2$ with radius r, I assume $r\le 1$ to avoid the pole at $s=1$, and perhaps the minimum disc radius is $r>\epsilon>0$.
Question 1: Is it true $\zeta(s)$ is completely determined for all $s\ne0$ by its values in a small open disc, and why the exclusion at $s=0$?
Question 2: Assuming the answer to question 1 is yes, can anyone provide insight as to how the value of $\zeta(s)$ at points outside of the disc are determined by the values of $\zeta(s)$ at points inside the disc?
I'm aware of the Cauchy formula where an integral along a contour enclosing a point $s_0$ is used to determine the value of $\zeta\left(s_0\right)$, but this question is for a very different case.