# Something related to Dedekind zeta function

Let $$K$$ be a number field, $$\mathcal{O}_K$$ its ring of integers, $$\mathcal{O}_K^{\times}$$ its group of units, $$\mathcal{Cl}(K)=J_K/P_K$$ its ideal class group. As you know, the Dedekind zeta function is defined as follows:

$$\zeta_{K}(s)=\sum_{I\subseteq \mathcal{O}_K}{\frac{1}{(N_{K/\mathbb{Q} }(I))^{s}}}.$$

Let $$\mathcal{A} \in \mathcal{Cl}(K)$$ be arbitrary, now consider this function:

$$\zeta_{K, \mathcal{A}}^{\text{BAD}}(s)=\sum_{I\subseteq \mathcal{O}_K \\ I \in [\mathcal{A}]}{\frac{1}{(N_{K/\mathbb{Q} }(I))^{s}}}.$$

I did not verify the conditions for absolute convergence, but if there is no serious problem about that, then we have:

$$\zeta_{K}(s) = \sum_{\mathcal{A} \in \mathcal{Cl}(K)} \zeta_{K, \mathcal{A}}^{\text{BAD}}(s).$$

$$\color{Red}{\text{My question}}$$:

I am very curious to know if the BAD zeta function $$\zeta_{K, \mathcal{A}}^{\text{BAD}}(s)$$, is a good thing or not? What other objects are related to the BAD zeta function $$\zeta_{K, \mathcal{A}}^{\text{BAD}}(s)$$? I have very little information about this subject, and I could not find anything about it in the literature. Is there something related to the BAD zeta function $$\zeta_{K, \mathcal{A}}^{\text{BAD}}(s)$$? Could you please introduce me to some references to follow and read?

• What does "$I\subseteq\mathcal{O}^\times_K$" mean? Jun 13, 2020 at 9:34
• @AnginaSeng +1, you are right. It doesn't make sense. I will edit it now. Jun 13, 2020 at 9:39
• Why ${}$BAD? Jun 13, 2020 at 9:53
• Your functions are "partial zeta functions". The standard proofs of the analytic class number formula involve using your final sum, and observing that each partial zeta function has a pole at $s=1$ with the same residue. Jun 13, 2020 at 10:05
• See my answer there math.stackexchange.com/a/3713036/276986 we decompose $\zeta_K$ this way because the sum over one ideal class is something closeto the Mellin transform of a theta function, from which we obtain the meromorphic continuation, the residues, the functional equation.. Jun 13, 2020 at 18:32