The way the class number formula for Dirichlet L-functions is always proved in modern textbooks and notes is

  1. You prove the general class number formula for $\zeta_K(s)$.

  2. You prove that for quadratic fields, $\zeta_K(s)=\zeta(s)L(s,\chi)$.

  3. Use 1. and 2. to evaluate $L(1,\chi)$.

Dirichlet proved the same formula for $L(1,\chi)$ before Dedekind proved the formula for $\zeta_K(s)$. I hear the he used the theory of quadratic forms.

Is there a proof of the formula for $L(1,\chi)$ in the language of ideals which doesn't use the class number formula for $\zeta_K(1)$ or the theory of quadratic forms?

  • $\begingroup$ you looked at terrytao.wordpress.com/2014/11/28/… ? $\endgroup$ – reuns Aug 23 '16 at 0:10
  • $\begingroup$ Dirichlet wrote a textbook, that has been translated into English, and is not too hard to read. Probably it follows his original proof closely. $\endgroup$ – Kimball Aug 23 '16 at 10:45

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