This is a simple question about manipulating power series, I can't find my error or maybe I just haven't finished it.
Let $X$ be a smooth projective variety over a finite field $K$. Let $C_n$ be the number of closed points $x$ of $X$ having residue field of degree $n$ over $K$ [EDIT: this was the source of my confusion. $C_n$ is actually supposed to be the number of $K_n$-rational points of $X$, where $K_n$ is the degree $n$ extension of $K$]. Let $$Z_X(t) = \prod_{x \in |X|, \ x \text{ closed}} \frac{1}{1-t^{\deg(x)}}$$
I want to show $$\log Z_X(t) = \sum_{n=1}^\infty \frac{C_n}{n} t^n$$
We have $\log(1-x)=-\sum_{i=1}^\infty \frac{t^i}{i}$.
So \begin{align*} \log Z_X(t) & = -\sum_{x \in |X|, \ x \text{ closed}} \log(1 -t^{\deg(x)}) \\ & = -\sum_{n=1}^\infty \left(\sum_{\deg(x)=n} \log(1 -t^{n})\right) \\ & = -\sum_{n=1}^\infty C_n\log(1 -t^{n}) \\ & = \sum_{n=1}^\infty \sum_{i=1}^\infty C_n\frac{t^{ni}}{i} \\ \end{align*} which doesn't seem to equal what it is supposed to equal.
This equals (setting $d=ni$) $$\sum_{d=1}^\infty\frac{t^d}d\sum_{n\mid d}n C_n.$$ Now, each point with residue field $\Bbb F_{q^n}$, for $n\mid q$ accounts for $n$ points on the curve defined over $\Bbb F_{q^d}$, so the coefficient $\sum_{n\mid d}n C_n$ is the total number of points on the curve defined over $\Bbb F_{q^d}$, exactly as it should be,