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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.

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    $\begingroup$ Actually $Z'(X,t)/Z(X,t)$ is not $P(X,t)$ but $P(X,t)/t$. $\endgroup$
    – GH from MO
    Sep 15, 2014 at 5:14
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    $\begingroup$ For curves, the rationality of the zeta function follows easily from the expression of the zeta function in terms of divisors and Riemann-Roch. But I don't see how to get the rationality of $P$ directly for curves. $\endgroup$ Sep 15, 2014 at 11:38

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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.

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  • $\begingroup$ That's awesome! Thinking about it now, it sound similar to what Terry Tao does in his paper on polynomial expalnders:arxiv.org/abs/1211.2894. $\endgroup$
    – jacob
    Sep 16, 2014 at 0:39

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