If $K$ is a number field with signature $(r_1, r_2)$, discriminant $D$, regulator $R$, class number $h$, containing $w$ roots of unity, and Dedekind $\zeta$-function $\zeta_K$, then $\zeta_K$ has a meromorphic continuation to the whole complex plane with a single pole at $s=1$, which is of order $1$. The **analytic class number formula** gives the residue at this pole:
$$ \lim_{s\to 1}\ (s-1)\zeta_K(s) = \frac{2^{r_1} (2\pi)^{r_2} R\cdot h}{w\sqrt{|D|}} .$$

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- Last edited by John Jones on 2020-10-12 09:26:47

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