We have that $\frac{X}{spec \mathbb{Z}}$, a scheme of finite type. Consider $\zeta(X,s)= \prod_{x\in{X}} \frac{1}{(1-Nx^{-s}}$, with $Nx$ the norm. I didn't catch what my teacher was saying and he is too busy to answer my question. He mentioned that $\zeta(A_\mathbb{Z}^n, s)=\zeta(s-n)$ and that this is a meromorphic continuation to all of $\mathbb{C}$. What exactly does this mean and how does it differ from an analytic continuation.
We have a theorem that says that $\zeta(X,s)$ has a meromorphic continuation to $\mathbb{R}e(s)>dim(X)-\frac{1}{2}$. I am not sure what this theorem is saying. Can someone help?
If $f$ is meromorphic on some connected open subset $U$ of the complex plane, and $V\supseteq U$ is another connected open subset, we say that $f$ has a meromorphic continuation to $V$ if there exists a meromorphic function $g$ on $V$ such that $g|_U = f$. The function $g$ is necessarily unique if it exists.