Let $X$ a curve over the field $\mathbb{F_{q}}$ (proper, integral normal scheme of dimension 1). The Riemann Zeta function for this curve is defined as
$$\zeta(X,t)=\prod_{x\ \text{closed}}\ \left(1-t^{\deg(x)}\right)^{-1}$$
where $\deg(x)=[k(x):k]$, $k(x)$ is the residue field and $t=q^{-s}$, $s$ complex number. I want to make this product precise. This means that I would like to show it belongs to $\mathbb{Z}[[t]]$, the formal power series. I know that actually the invertible element $\left(1-t^{\deg(x)}\right)^{-1}$ is in $\mathbb{Z}[[t]]$ equal to
$$\left(1-t^{\deg(x)}\right)^{-1}=\sum_{i=0}^{\infty}t^{i\deg(x)}$$ So, this is a well defined element in $\mathbb{Z}[[t]]$. I also know the fact that the number of closed points $x$ such that $\deg(x)\leq N$ is finite, where $N\in\mathbb{N}$. But I cannot use it to express the Zeta function as an element in $\mathbb{Z}[[t]]$.