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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?

Davood
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    What does "$I\subseteq\mathcal{O}^\times_K$" mean? – Angina Seng Jun 13 '20 at 09:34
  • @AnginaSeng +1, you are right. It doesn't make sense. I will edit it now. – Davood Jun 13 '20 at 09:39
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    Why ${}$BAD? – Angina Seng Jun 13 '20 at 09:53
  • @AnginaSeng because I don't have enough courage to speak about it, and I thought maybe that function is ridiculous for an expert. – Davood Jun 13 '20 at 09:59
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    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. – Angina Seng Jun 13 '20 at 10:05
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    See my answer there https://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.. – reuns Jun 13 '20 at 18:32

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