Let \(K\) be a number field and let \(\mathcal{O}_{K}\) be its ring of integers. \(\Re(s)>1\), we define the Dedekind-zeta function to be \[\begin{align} \zeta_K(s) = \sum_{\substack{ \mathcal{I} \subseteq \mathcal{O}_{K} \\ \mathcal{I} \text{ non-zero ideal} } } \frac{1}{\mathop{\mathrm{N}}(\mathcal{I})^{s}}, \end{align}\] where \(\mathop{\mathrm{N}}(\mathcal{I}) = \# \left( \mathcal{O}_{K} / \mathcal{I}\right)\).
For \(\Re(s) > 1\), we have the following product expansion.
\[\begin{align} \zeta_K(s) = \prod_{\substack{ \mathcal{P} \subseteq \mathcal{O}_{K} \\ \mathcal{P} \text{ prime ideal} } } \frac{1}{\left( 1 - \frac{1}{\mathop{\mathrm{N}}(\mathcal{P})^{s}} \right)}. \end{align}\]
For cyclotomic fields, one can evaluate \(\zeta_K(s)\) more precisely.
See Dedekind-zeta for cyclotomic fields
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This page was updated on November 13, 2023.
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