Why are the Weil Conjectures stated via zeta functions? Let $C$ be a smooth curve over $\mathbf{F}_p$, and let $\zeta$ be the zeta function of $C$. The Weil Conjectures for $C$ are usually stated something like this:

*

*The zeta function $\zeta(s)$ is a rational function of $p^{-s}$.

*The zeta function satsfies a functional equation: $\zeta(s) = \zeta(1-s)$.

*The zeroes of $\zeta(s)$ lie on the line $\text{Re }s = 1/2$.

I understand how these theorems imply strong bounds on $\#C(\mathbf{F}_{p^n})$. So for example, they imply that if $C$ has genus $g$, then:
$$|\#C(\mathbf{F}_{p^n}) - (p^n+1) | \leq 2g \sqrt{p^n}.$$
My question is: if the results of the Weil conjectures can be translated into giving bounds for $\#C(\mathbf{F}_{p^n})$ for various $n$ (like the inequality above), then why do we bother stating the Weil Conjectures via zeta functions? Why not just cut the middleman and state them in terms of the bounds they imply on $\#C(\mathbf{F}_{p^n})$? Is there an advantage to stating them first in terms of zeta functions and then proving these inequalities as a consequence? What additional insight does packaging this information into this zeta function give about the underlying math?
To be clear, I am not asking for motivation for the definition of zeta; I understand where the literal definition of $\zeta$ comes from. Rather, I am asking: why do we want to take the various $\# C(\mathbf{F}_{p^n})$ and package them into a zeta function in the first place? What is the utility of this? I'm sure there are very good reasons, so any suggestions would be more than welcome. Thanks!
 A: Various zeta functions are basically generating functions for the number of solutions. Generating functions are extremely valuable as they embed information about varieties. Many conjectures are easily translated into the language of zeta functions. By expressing them that way, the solution of some problem is either giving more insights about the generating function itself or giving a better tool to handle a generating function of similar nature.
Your question is then: why do we want to deal with generating functions?
If you want to deal with various questions about convergence of a series, it is best if you can skip into the complex realm, because this one is going to give you almost a complete insight over how your series behaves, in a way that is virtually impossible without going into complex analysis. It simply echoes so many details that you can say that you know everything that is to know about convergence for quite some families of series.
Generating functions, especially when defined in the complex domain, are so rich source of information that we want to use them to express our current knowledge boundaries as we expect to deal with these objects for many more decades before we can say that we have something spectacularly new. Those that gave in already are confirming that more you are able to deal with them more you know about a specific object to the extent that you regularly claim that two objects are isomorphic in some sense if their generating functions match.
When you conjecture something about the number of solutions, it is best if you can express it within the zeta function as your insight cuts deeper as it immediately branches into a very wide number of mathematical branches. Your findings become instantly translated into very different mathematical languages and each author can argue about the impact in his area of expertise.
All that through one device only.
In that sense, zeta functions are (no longer) observed as something that comes from the number of solutions, for example, that is just one of the packages of information it contains. We observe them more as an object on their own which contains, if not everything, then a very valuable piece of information about the variety in question.
Generating function are at the same time the easiest and the most difficult object to deal with. The easiest as we have expressed quite some number of mathematical facts using these, so we are able to deal with generating functions in technical sense; most difficult as some facts look so elementary when expressed through generating functions and yet we have no clue whatsoever where to start, and this goes on for decades.
There is something profoundly strange within the realm of zeta functions as our ignorance is so well embedded in them. When you express it that way, you know exactly how little you do know. At least that much is good to have anyway.
