TITLE: Is it easy to prove that $\sum_n |X(\mathbb{F}_{q^n})| t^n$ is rational?
QUESTION [12 upvotes]: Background: Let $X$ be an algebraic variety over a finite field $\mathbb{F}_q$. One  of the successes of Etale cohomology - previously achieved by Dwork-  was proving the rationality of the Zeta function of X, defined as 
$$Z(X,t)=\textrm{exp}\left(\sum_{n\geq 1}\frac{|X(\mathbb{F}_{q^n})|t^n}{n}\right).$$
As a corollary, we get the rationality of $tZ'(X,t)/Z(X,t)$, which is the sum 
$P(X,t)=\sum_n |X(\mathbb{F}_{q^n})| t^n.$ 
Question: Is there an elementary way to see that $P(x,t)$ is a rational function? Note that this is strictly weaker than asking whether $Z(X,t)$ is a rational function.

REPLY [11 votes]: Not an answer to the question: in a 1976 Transactions of the AMS paper, Catarina Kiefe shows that the logarithmic derivative of zeta is rational for sets definable over finite fields (it is NOT necessarily true that zeta itself is, as she points out in the paper). So, the good news is that the rationality of $Z^\prime/Z$ is strictly weaker than rationality of $Z,$ the bad news is that she uses Dwork's result (and a bunch of model theory, though the paper is quite short). The fact that she makes no reference to a direct proof of such a result for varieties may be a sign that none was known at the time.