In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. General statement of the class number formula We start with the following data: is a number field. , where denotes the number of real embeddings of , and is the number of complex embeddings of . is the Dedekind zeta function of . is the class number, the number of elements in the ideal class group of . is the regulator of . is the number of roots of unity contained in . is the discriminant of the extension . Then: Theorem (Class Number Formula). converges absolutely for and extends to a meromorphic function defined for all complex with only one simple pole at , with residue This is the most general "class number formula". In particular cases, for example when is a cyclotomic extension of , there are particular and more refined class number formulas. Proof 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. Dirichlet class number formula Peter Gustav Lejeune Dirichlet published a proof of the