In the paper Zum Beweise des Starkschen Satzes Siegel considers the function


where $q$ is a discriminant of a quadratic number field and the character is the Kronecker symbol. Then he writes that "according to Dirichlet" we have, in case $G>0$,

$$L_G(1)=2G^{-1/2}h_G\log \varepsilon_G,$$

where $h_G$ is the corresponding class number and $\varepsilon_G$ the fundamental unit.

However, according to the book Zetafunktionen und quadratische Körper the formula reads

$$h_G=\frac{G^{1/2}}{\log \varepsilon_G}L_G(1).$$

Which one is correct? Am I making some mistake?

  • $\begingroup$ you mean where does the factor of two in the first formula come from? $\endgroup$ – Carlo Beenakker Mar 21 '20 at 16:08
  • $\begingroup$ @CarloBeenakker Yes, that is the problem. $\endgroup$ – Shimrod Mar 21 '20 at 16:08
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    $\begingroup$ Dirichlet's class number formula is quoted with this factor of two in several other sources, for example here. $\endgroup$ – Carlo Beenakker Mar 21 '20 at 16:27
  • $\begingroup$ Maybe it would be wiser to read Dirichlet's proof itself. $\endgroup$ – Sylvain JULIEN Mar 21 '20 at 16:52
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    $\begingroup$ Siegel is always right. $\endgroup$ – GH from MO Mar 23 '20 at 0:16

Dirichlet proved his class number formula for quadratic forms; in particular he was working with class numbers $h^+$ in the strict sense, and his unit $\varepsilon$ was the fundamental solution of the Pell equation $t^2 - Du^2 = 1$, not the fundamental unit $\eta$ of the corresponding number field. The relation $$ \eta^{2h} = \varepsilon^{h^+} $$ encodes the two cases

  • $\varepsilon = \eta$, $h^+ = 2h$
  • $\varepsilon = \eta^2$, $h^+ = h$.
  • $\begingroup$ Thank you for your answer! $\endgroup$ – Shimrod Mar 24 '20 at 16:30

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