I was reading through chapter 16 of Lang's Algebraic Number Theory book. The chapter is fully devoted to proving the Brauer-Siegel theorem: Let ${(k_n/ \mathbb{Q})}_n$ be a sequence of galois extensions over $\mathbb{Q}$ of degree $N_n$, such that $\frac{N_n}{log(d_n)} \to 0$ then $log(h_n R_n) \sim \frac{1}{2}log(d_n)$, where $d_n, h_n, R_n$ are the absolute value of the disciminant, class number and regulator of $k_n$ respectively. The following is one of the main theorems in the chapter.

As said in the proof, the theorem easily follows if RH (or rather GRH) is assumed from a previous lemma which is as follows,

As given in the lemma it is only required to assume that the Dedekind zeta function $\zeta_k(s_0) \leq 0$. He treats case 2, separately going through several other lemmas.

Now in case 2 picking $s_0 \in (1 - \frac{\epsilon}{N}, 1)$ a zero of $\zeta_k(s)$ we would still have that $\zeta_k(s_0) \leq 0$ and the result immediately follows from the lemma as in case 1. So why do we have to treat it separately?

Edit:

Going through the separate treatment of case 2 shown below, I understand that using the fact that $\zeta_{k_0}(s_0) = 0$, the theorem is proven for other number fields that may have the same pathological behaviour. Let me know if this line of thought is correct. Well, then Theorem 1 should have been "... for all fields normal over $\mathbb{Q}$ except for one possible exception..." since the theorem is left unproven for $k_0$.

Edit: Here $\kappa(k) = \frac{2^{r_1}{(2\pi)}^{r_2}h_k R_k}{\omega {d_k}^{\frac{1}{2}}}$ is the residue of $\zeta_k(s)$ at $s = 1$.

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