Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K\_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be zero.


Since the paper is from 2004 I was wondering if anything new is known about those groups.


I'm interested on any new research on them.



References for $K_{4k}(\mathbb{Z})$
Well, the consensus seems to be that this is an open problem, for $k >1$.


This is a quote from A. Raghuram's paper on the volume "[The Bloch–Kato Conjecture for the Riemann Zeta Function](http://www.cambridge.org/us/academic/subjects/mathematics/number-theory/blochkato-conjecture-riemann-zeta-function?format=PB#bookPeople)" (page 8), published in april 2015.



> 
> [...] it is expected $K\_{4a}(\mathbb{Z})=0$. This is proven for $a=1$
>  and is open as yet for $a\geq 2$.
> 
> 
> 


Given that other contributors include Stephen Lichtenbaum and Manfred Kolster, this most likely represents the current knowledge at least at the time of the conference (2012).


If someone has any update (particularly regarding Philippe Elbaz-Vincent's claim, see the comments above), feel free to answer or comment.


