Is there a unified description (or a set of axioms) of the zeta function of an algebraic curve over a finite field $\mathbb{F}\_q$ and the Riemann zeta function? 



A unified description of zeta functions of a curve over $\mathbb{F}_q$ and Riemann $\zeta$ function
Let $D$ be a ring which is infinite, not a field, and in which $D/I$ is finite for any nonzero ideal $I$ (what Pete Clark calls a [abstract number ring](http://alpha.math.uga.edu/~pete/aant.pdf)). Then we can define $N(I) = |D/I|$ and write down a zeta function
$$\zeta\_D(s) = \sum\_{I \neq 0} \frac{1}{N(I)^s}.$$


This specializes to the Riemann zeta function when $D = \mathbb{Z}$, to the Dedekind zeta function when $D = \mathcal{O}\_K$, and to the zeta function of an affine algebraic curve over a finite field when $D$ is its ring of functions. If $D$ is Dedekind, then unique factorization of ideals gives an Euler product for this zeta function.


To get the zeta function of a curve, not necessarily affine, we replace ideals with effective divisors, that is, non-negative formal integer linear combinations
$$\sum n\_i P\_i, n\_i \ge 0$$


of points of $C$ over $\overline{\mathbb{F}\_q}$ which are $\text{Gal}(\overline{\mathbb{F}\_q}/\mathbb{F}\_q)$-invariant, and the appropriate replacement for the norm here is $q^{\sum n\_i}$. This reduces to the above description in the case of a (nonsingular, geometrically integral) affine curve. (The more standard expression for the zeta function of a curve over a finite field, which tells you what the logarithm of the zeta function is in terms of counting points over finite extensions of $\mathbb{F}\_q$, can be shown to be equivalent to this description using the exponential formula in combinatorics. See [this blog post](http://qchu.wordpress.com/2009/11/04/newtons-sums-necklace-congruences-and-zeta-functions-ii/).)


