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Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\zeta_Q(s) = \sum_{(x,y)\in \mathbb{Z}^2-0} \frac{1}{Q(x,y)^s}$$ is essentially partial zeta series $\zeta_K(I,s)$ for an ideal class of $K$, so $\zeta_Q(s)$ is a linear combination of Artin L-function for characters of $\text{Gal}(L/K)$, here $L$ is Hilbert class field of $K$.


If $D$ is not fundamental, let $\mathcal{O}$ be the associated non-maximal order. I think $\zeta_Q(s)$ is still of the following form $$\tag{*}\zeta_Q(s) = \sum_i f_i(s) L(\chi_i,s)$$ here $f_i(s)$ is a certain finite Euler product, and $L(\chi_i,s)$ are Artin L-function with $\chi_i$ character of $\text{Gal}(L/K)$, with $L$ the ring class field of $\mathcal{O}$.

Question: is $(*)$ true in general? how to explicitly find the $f_i(s)$?

I think an explicit form of $f_i$ would likely to be complicated, so proving their existence is already nice enough.


As an example of $(*)$: let $Q(x,y) = x^2 + 4y^2, K = \mathbb{Q}(\sqrt{-1}), \mathcal{O} = \mathbb{Z}[2\sqrt{-1}]$, ring class field $L$ is same as $K$, so there is only one $L(\chi_i,s) = \zeta_K(s)$, we have $$\sum_{(x,y)\in \mathbb{Z}^2-0} \frac{1}{(x^2+4y^2)^s} = 2 \left(1-2^{-s}+2\times 4^{-s}\right) \zeta_K(s)$$

K. Williams and others theorem 10.1 of this paper have proved that $(*)$ is true if the class group of $\mathcal{O}$ is $2$-torsion; more generally, $(*)$ is true if we sum over a genus.

Any idea is welcomed, thank you.

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    $\begingroup$ Please use a high-level tag like "nt.number-theory". I added this tag now. $\endgroup$
    – GH from MO
    Commented May 25, 2023 at 16:17

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