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If $\mathcal{O}$ is the ring of integers of a number field, then the Hecke-L-series for a character $\chi$ of the class group is defined as $$L(\chi,s) = \sum_{\mathfrak{a} \neq 0}\frac{\chi(\mathfrak{a})}{N(\mathfrak{a})^s}$$ with $N(\mathfrak{a}) = (\mathcal{O}:\mathfrak{a})$. Using this L-series one can proof the generalized Dirichlet density theorem (generalizes Dirichlet's theorem on primes in arithmetic progressions to primes in certain ideal classes) of whom the celebrated Cebotarev density theorem is a corollary.

Now let $D$ be a Dedekind domain $D$ with the property that $D/P$ is finite for each prime $P \subseteq D$. Then one can obviously define an L-series as above.

Question: Are there examples of Dedekind domains where L-series were used to prove similar density theorems as in the case of number fields ?

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You say that one can obviously define an $L$-series in the setting of your abstract $D$. Please tell us your obvious definition of a Hecke character on (nonzero) ideals in $D$. –  KConrad Feb 20 '12 at 1:18
In order to generalize Dirichlet's theorem you need to consider characters of ray class groups. As it stands, all your $L$-functions have trivial conductor. –  GH from MO Feb 20 '12 at 1:43
KConrad, GH: Thanks for pointing out the problems associated with the charcters. I'll have a closer look on their properties in the number field case. –  Ralph Feb 20 '12 at 7:17

1 Answer 1

A classical example would be the 'geometric' analogs. So, holomorphy rings of algebraic function fields over finite fields. Or, to give a special case of this that almost does not qualify (being a UFD): a polynomial ring in one variable over a finite field.

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quid, thanks for the examples. –  Ralph Feb 20 '12 at 7:34

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