The Analytic Class Number Formula

A handwavy introduction to Tate’s Thesis

Andrei Jorza March 14, 2005

1 The ζ-function

Let K be a number field. Definition 1.1. The ζ-function is

ζ_{K}(s) = ∑_{I} \frac{1}{NI^{s}} ,

where I runs through all the ideals of O_{K}. Proposition 1.2. We have

ζ_{K}(s) = ∏_{𝔭∈Spec O_{K}} \frac{1}{1 − N𝔭^{−s}} .

Proof. There are sensible convergence issues here, but we will not worry about these. Since O_{K} is a Dedekind domain, with unique factorization, every ideal I = 𝔭_{1}^{n_{1}} ⋯ 𝔭_{k}^{n_{k}} so

ζ_{K}(s) = ∑(N𝔭_{1}^{n_{1}} ⋯ 𝔭_{k}^{n_{k}})^{−s} = ∏_{𝔭_{i}} ∑_{n_{i}=0}^{∞}(N𝔭_{i})^{−n_{i}s} = ∏_{𝔭_{i}} \frac{1}{1 − N𝔭_{i}^{−s}} .

The ζ_{K}(s) clearly converges for Res > 1.