From my previous [question](https://mathoverflow.net/questions/207885/reference-request-zeta-function-is-rational-function-via-riemann-roch), I know that$$\zeta\_X(s) = {{P(u)}\over{(1-u)(1-qu)}}$$for some polynomial $P(u)$ of degree $2g$, where


* $X$ is the set of all places of $F$, a function field in one variable over a total constant field $k$,
* $u := q^{-s}$,
* $g$ is the genus of $F$.


My question now is, must the limit$$\lim\_{s \to 1} (1 - q^{1-s})\zeta\_X(s)$$necessarily converge to$${h\over{(q-1)q^{g-1}}}?$$Here, $h$ denotes the number of divisor classes of $F$ of degree $0$.



Question about zeta function of function field in 1 variable over $\mathbb{F}_q$
Yes. From the functional equation (Theorem 4 in Chapter VII-6 of Weil: Basic number theory) we know that $P(u)=q^gu^{2g}P((qu)^{-1})$. We also know that (see same theorem) that $P(1)=h$. Hence
the limit in question equals
$$ \frac{P(q^{-1})}{1-q^{-1}}=\frac{q^{-g}P(1)}{1-q^{-1}}=\frac{P(1)}{q^{g-1}(q-1)}=\frac{h}{q^{g-1}(q-1)}.$$


