Analytic class number formula

In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.

We start with the following data:

Then:

This is the most general "class number formula". In particular cases, for example when K is a cyclotomic extension of Q, there are particular and more refined class number formulas.

The idea of the proof of the class number formula is most easily seen when K = Q(i). In this case, the ring of integers in K is the Gaussian integers.

An elementary manipulation shows that the residue of the Dedekind zeta function at s = 1 is the average of the coefficients of the Dirichlet series representation of the Dedekind zeta function. The n-th coefficient of the Dirichlet series is essentially the number of representations of n as a sum of two squares of nonnegative integers. So one can compute the residue of the Dedekind zeta function at s = 1 by computing the average number of representations. As in the article on the Gauss circle problem, one can compute this by approximating the number of lattice points inside of a quarter circle centered at the origin, concluding that the residue is one quarter of pi.

The proof when K is an arbitrary imaginary quadratic number field is very similar.

In the general case, by Dirichlet's unit theorem, the group of units in the ring of integers of K is infinite. One can nevertheless reduce the computation of the residue to a lattice point counting problem using the classical theory of real and complex embeddings and approximate the number of lattice points in a region by the volume of the region, to complete the proof.

Peter Gustav Lejeune Dirichlet published a proof of the class number formula for quadratic fields in 1839, but it was stated in the language of quadratic forms rather than classes of ideals. It appears that Gauss already knew this formula in 1801.

...
Wikipedia