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](https://link.springer.com/referenceworkentry/10.1007/978-3-540-27855-9_9)):


$$\zeta\_{K}(1-n) = \pm 2^{r\_1} \frac{|K\_{2n−2}(O\_K)|}{|K\_{2n−1}(O\_K)|} = \pm \frac{|H^{2}(O\_K;Z(n))|}{|H^1(O\_K;Z(n))|}.$$


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](http://users.ictp.it/~pub_off/lectures/lns015/Kolster/Kolster_Final.pdf). 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 $K=\mathbb{Q}$.



Status of the extended form of the Lichtenbaum conjecture
Thanks very much for the question!


Looking it up a bit I found the PhD thesis "[The Lichtenbaum Conjecture at the Prime 2](https://macsphere.mcmaster.ca/handle/11375/14321)
" 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.


