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?

the multiplicative functionsand those for which $f(\mathfrak{p})$ can be defined without fixing an enumeration of the primes). $\endgroup$ – reuns Aug 11 at 12:37complex-valuedzeta and $L$-functions of characteristic $p$ objects andcharacteristic-$p$-valuedzeta and $L$-functions of characteristic $p$ objects. For the first kind see Mike Rosen's "Number Theory in Functions Fields" and for the second kind see David Goss's "Basic Structures of Function Field Arithmetic" (or see Section 3.1 of people.math.osu.edu/goss.3/zeroes.pdf). $\endgroup$ – KConrad Aug 11 at 13:50