Let $X=A^n$. I am consider generating series $P_X(t)=\sum_{r\geq 1}v_rt^r$ where $v_r$ is number of $F_{p^r}$ points in $X$. I knew $P_X(t)=\frac{Z'_X(t)}{Z_X(t)}t$ where $Z_X(t)$ is the zeta function of $X$ defined in Shafarevich.
Since $X=A^n$, all $(F_{p^r})^n$ points are counted. So I have $v_r=p^r$. So $P_X(t)=\frac{1}{1-pt}$ formally. Now I wish to solve for $Z_X(t)$. Since $Z_X(0)=1$ by definition. I have $log(Z_X(t))=log\frac{t}{1-pt}+C$ after integration by parts. However, I cannot plug in $t=0$ as this clearly does not make any sense in log part.
Q: What is wrong in the argument above? Maybe my first step of identification does not make any sense? I could assume $|t|<1$.
The sum for $P_X(t)$ starts at $r=1$ rather than $r=0$, so $P_X(t)=pt/(1-pt)$. You find $\log(Z_X(t))$ by integrating $P_X(t)/t$, so using the new formula for $P_X(t)$ gives $\log(Z_X(t)) = -\log(1-pt)$. The constant of integration must be $0$ because $Z_X(0)=1$.