Status of the extended form of the Lichtenbaum conjecture The extended Lichtenbaum conjecture concerns the relationship of special values of $L$-functions of number fields $K$, to the algebraic $K$-theory and etale cohomology of the ring of integers $O_K$.
For example, when $K$ is totally real and $n$ is even, it is known that (cf. Section 4.7.4 of Kahn):
$$\zeta_{K}(1-n) = \pm 2^{r_1}  \frac{|K_{2n−2}(O_K)|}{|K_{2n−1}(O_K)|} = \pm \frac{|H^{2}(O_K;Z(n))|}{|H^1(O_K;Z(n))|}.$$
I am interested in the case $n$ odd. In that case, the conjectured value involves additionally a determinant of values of the Beilinson regulator. In particular, formula (4.34) of the aforementioned reference says that the conjectured value "should" be 
$$\lim_{s \to 1−n} (s+n)^{-g_n} \zeta_K(s) = \pm \frac{|H^{2}(O_K;Z(n))|}{|H^{1}(O_K;Z(n))_{tors}|} R'_n(K).$$
For certain number fields $K$, this is known to be true up to a power of 2, e.g. Theorem 4.1 of Kolster. However, this and other references go back a number of years, and it seems not impossible to me that nowadays more is known:

For which number fields $K$ is the above formula for $\lim_{s \to 1−n} (s+n)^{-g_n} \zeta_K(s)$ known to be correct, without discarding powers of 2?

I am particularly interested in the case $K=\mathbb{Q}$.
 A: Thanks very much for the question!
Looking it up a bit I found the PhD thesis "The Lichtenbaum Conjecture at the Prime 2
" by Ion Rada (a student of Kolster) which proves that for every abelian number field $K$ and every odd integer $n \geq 3$ one has
$$ \zeta_K^{\ast}(1 - n) = \varepsilon_n(K) \cdot 2^{\mu_2(K^{+})} \cdot \frac{h_n(K)}{w_n(K)} \cdot R_n^{B}(K) $$
where:


*

*$\varepsilon_n(K) \in \{ \pm 1\}$ is determined by the $\Gamma$-factors in the functional equation;

*$h_n(K) := \prod_p \lvert H^2_{\text{et}}(\mathcal{O}_K^{S_p(K)},\mathbb{Z}_p(n)) \rvert$, where (as far as I understand) $S_p(K)$ is the set of places of $K$ consisting of the places lying above $p$ and the Archimedean places corresponding to real embeddings;

*$w_n(K) := \prod_p \, \lvert H^0(K,\mathbb{Q}_p/\mathbb{Z}_p(n))\rvert $;

*$R_n^B(K)$ denotes Beilinson's regulator;

*$K^{+}$ is the maximal real subfield of $K$;

*for every number field $F$ and every prime $p$ we set $\mu_p(F)$ to be the $\mu$-invariant of the module $U_p(F)/C_p(F)$, where $U_p(F)$ denotes the projective limits of the groups $\mathcal{O}_{F_{p^r}}^{\times}$, where $F_{p^r}$ denotes the $r$-th field in the cyclotomic $\mathbb{Z}_p$-tower of $F$, and $C_p(F)$ denotes the group of circular units.


I believe that the main ingredient is the proof of the Milnor conjecture by Voevodsky.
