Reference request for Euler products in positive characteristic

Question

Let $K$ be a global field with ring of integers $O$, and let $f$ some integer valued function whose domain is the set of ideals of $O$ (e.g. $f(I)=|O:I|$). Extending the ordinary definition from the case $K=\mathbb Q$, one says that $f$ is multiplicative if $\newcommand{\mfp}{\mathfrak{p}}f(\mfp_1\mfp_2)=f(\mfp_1)f(\mfp_2)$ for any two relatively prime primes $\mfp_1,\mfp_2\triangleleft O$. In particular, if $f$ is multiplicative and positive then $f(0)=1$.

One can define a Dirichlet function associated to $f$: $$\zeta_f(s)=\sum_{I\triangleleft O} f(I)^{-s}\quad(s\in C) $$ and, assuming the domain of convergence is non-empty, obtain an Euler type factorization: $$\zeta_f(s)=\prod_{\mfp\in \mathrm{Spec}(O)\setminus 0}\zeta_{f,\mfp}(s)$$ where $\zeta_{f,\mfp}(s)=\sum_{n\ge 0}f(\mfp^n)^{-ns}$.

In the case where $\mathrm{char}(K)=0$ there's a rather well-established theory for analysing such products, including an interpretation of its abscissa of convergence (the infimal real number bounding the domain of convergence) in terms of the growth rate of the partial sums $$\sum_{I\triangleleft O,\:|O:I|<N}f(I),$$ as well as analytic results giving estimates for this sum in the case some meromorphic continuation of $\zeta_f$ exists.

Is anybody aware of any parallel results for the case $\mathrm{char}(K)>0$? Are there any common references regarding these types of Euler factorization in positive characteristic? For example- is the Dedekind zeta function of a function field a studied object, or are there obvious restrictions why one should not attempt to study such a function?

Answer 1

It's better to just work with effective divisors. An effective divisor is simply a formal sum with nonnegative integer coefficients of finitely many valuations of $K$ (= closed points of the curve that $K$ is the function field of). The prime ideals of any ring of integers of $K$ will be naturally in bijection with this set of valuations, minus finitely many.

A multiplicative function is a function on effective divisors $D$ with $f(0)=1$ and $f(D_1+D_2) = f(D_1)f(D_2)$ as long as $\operatorname{supp}(D_1) \cap \operatorname{supp}(D_2)=\emptyset$. For $f$ multiplicative, we clearly have

$$ \sum_{D \geq 0} f(D)q^{ - (\operatorname{deg} D)s } = \prod_{ v} \sum_{n\geq 0} f( n[v]) q^{ - n (\operatorname{deg} v ) s } .$$

Here the degree of a valuation is the degree of its residue field over $\mathbb F_q$, and the degree of a divisor is the sum with nonnegative integer coefficients of the degrees of its valuations. $q^{ \operatorname{deg} D}$ is the analogue of $|D|$. I have no idea why you have raised the multiplicative function $f$ to the power $-s$ (possibly because you don't know the definition of the norm of an ideal in the function field case?)

If $f$ is the constant function $1$, then this product is known as the Weil zeta function of the curve whose function field is $K$ and is well-studied. The Riemann hypothesis for it was proved by Weil (the meromorphic continuation was known prior). Many cases of Riemann hypothesis for the analogue of Hecke $L$-functions was also worked out by Weil, and Artin $L$-series are fine too, answering reun's question. However, it is certainly necessary to work with the usual definition of Dirichlet series and not raise $f$ to a power.

The abscissa of convergence step is easier in this setting. One just rewrites the sum $$ \sum_{D \geq 0} f(D)q^{ - (\operatorname{deg} D)s } = \sum_{d \geq 0}\left( \sum_{\substack {D \geq 0 \\ \deg D = d}} f(D) \right) q^{-ds} , $$ views this as a power series in $q^{-s}$, and uses standard results on the radius of convergence of this power series.