Class Number Formula - Dirichlet Class Number Formula

Dirichlet Class Number Formula

This exposition follows Davenport. The first class number formula was proved by Dirichlet in 1839, but it was proved about classes of quadratic forms rather than classes of ideals. Let d be a fundamental discriminant, and write h(d) for the number of equivalence classes of quadratic forms with discriminant d. Let be the Kronecker symbol. Then is a Dirichlet character. Write for the Dirichlet L-series based on . For d > 0, let t > 0, u > 0 be the solution to the Pell equation for which u is smallest, and write

(Then ε is either a fundamental unit of the real quadratic field or the square of a fundamental unit.) For d < 0, write w for the number of automorphs of quadratic forms of discriminant d; that is,

w = \begin{cases} 2, & d < -4; \\ 4, & d = -4; \\ 6, & d = -3. \end{cases}

Then Dirichlet showed that

h(d)= \begin{cases} \frac{w \sqrt{|d|}}{2 \pi} L(1,\chi), & d < 0; \\ \frac{\sqrt{d}}{\ln \epsilon} L(1,\chi), & d > 0. \end{cases}

This is a special case of Theorem 1 above: for a quadratic field K, the Dedekind zeta function is just, and the residue is . Dirichlet also showed that the L-series can be written in a finite form, which gives a finite form for the class number. We have

L(1, \chi) = \begin{cases} -\frac{\pi}{|d|^{3/2}}\sum_{m=1}^{|d|} m \left( \frac{d}{m} \right), & d < 0; \\ -\frac{1}{d^{1/2}}\sum_{m=1}^{d} \left( \frac{d}{m} \right) \ln \sin \frac{m\pi}{d}, & d > 0. \end{cases}

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