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Is there a way to find the partial sums of the Dedekind zeta function? In particular, I would like to find the sum $$\sum_{0 \neq I \subset\mathcal{O}_K \\N(I) \le n} \frac{1}{N(I)^s}$$ for a given $n$ and $s$?

Edit. For quadratic numbeer fields, by using the identity $\zeta_K(s)=\zeta(s)L(s,\chi)$ as here: Properties of Dedekind zeta function I can able to find the value of partial sums of $\zeta_K(s)$. For example, for cyclotomic number fields, the same method works too. However, I look for a function etc. defined in the computer algebra system.

Ninja
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    Find out how the prime ideals behave in your ring of integers. Then you know which ideals have norm small $n$ and you can sum it up. –  Mar 07 '20 at 10:28
  • What do you mean with "find" ? $\frac{\zeta_K(s+z)}{z}$ is the Mellin transform of $\sum_{0<N(I) \le x} N(I)^{-s}$. For a multiplicative function $a_n$ in general there is no easy way to compute $\sum_{n\le x} a_n$. – reuns Mar 07 '20 at 10:49
  • Okay, I mean for example in the case $s=1$, we can proceed as in https://math.stackexchange.com/questions/92426/how-many-elements-in-a-number-field-of-a-given-norm.

    What about $s=2,3, $etc.?

     – Ninja Apr 09 '20 at 12:35

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