I want to know if there is a good combinatorial interpretation of what the Zeta function of a variety $X$ over a finite field $\mathbb{F}_p$ counts. It is defined as $$\exp\sum N_j/j\,t^j,$$ where $N_j$ is the number of points in $X$ over $\mathbb{F}_{p^j}$.
I hope an answer will be something along the following lines: the coefficients of $\sum N_j/j\,t^j$ count the orbits of the solution sets under iterations of the Fröbenius automorphism, and the exponential of the sequence somehow counts representatives.
I reckon this is very elementary combinatorics. A concise source I could use to produce such a combinatorial statement would be very much appreciated!
The zeta function counts effective divisors, e.g. objects of the form $\sum n_i p_i$ where
If $X$ is affine, this is equivalent to saying that it counts ideals of the coordinate ring of $X$; in fact in this setting the zeta function is more or less the Dedekind zeta function of the coordinate ring.
This is a corollary of a version of the exponential formula which asserts that
$$\exp \sum \frac{z_j}{j} t^j = \sum_{n \ge 0} Z_{S_n}(z_1, z_2, ...) t^n$$
where $Z_{S_n}$ are the cycle index polynomials of the symmetric groups. See this blog post and this blog post for details.