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.