Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field.
How the following implication works: $$Z(t,C)=\frac{\prod_{i=1}^{2g}{(1-\alpha_{i}t)}}{(1-t)(1-qt)}$$ $\Rightarrow exp(\sum_{m=1}^{\infty}N_m \frac{t^m}{m})=exp(\sum_{m=1}^{\infty}(q^n + 1 - \sum_{i=1}^{2g}{a_{i}^n})\frac{t^n}{n})$
Should one take firstly the logarithmic derivative of $\frac{\prod_{i=1}^{2g}{(1-\alpha_{i}t)}}{(1-t)(1-qt)}$ and then integrate this and apply $exp(\cdot)$?
(similarly works for $Z(t,C)=exp(\sum_{m=1}^{\infty}N_m \frac{t^m}{m})$, where $N_m:=\sum_{P,d(P)|m}{d(P)}$ (see below)) I tried this approach, but I didn't get the desired.
Following is the context (see also C. Moreno, curves over finite fields):
Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field.
The zeta function of $C$ is defined by $Z(t,C):= \sum_{D} {t^{d(D)}} $, where the sum goes through all positive divisors.
One can show that
Theorem: $Z(t,C)=exp(\sum_{m=1}^{\infty}N_m \frac{t^m}{m})$, where $N_m:=\sum_{P,d(P)|m}{d(P)}$