rationalized K-Theory of the group ring of finite cyclic groups

Question

I am interested in calculating the rationalized algebraic K-Theory groups of the group ring of $\mathbb Z/n$, that is $K_i(\mathbb Z[\mathbb Z/n])\otimes \mathbb{Q}$ for any natural number $n\geq 2$. What is known about the rationalized K-Theory of $\mathbb Z[\mathbb Z/n]$ for an arbitrary $n$?

Answer 1

The rationalized K-theory of finite groups is known: Let $G$ be a finite group and $n \ge 2$. $$K_n(\mathbb Z G)\otimes \mathbb Q = \begin{cases} \mathbb{Q}^r & n\equiv 1(4) \newline \mathbb{Q}^c & n\equiv 3(4) \newline 0 & n \text{ even} \end{cases}$$ where $r$ is the number of irreducible real representations of $G$ and $c$ of them are of complex type (see Theorem 2.2 here).

Now let $G=\mathbb{Z}/n$. If $n$ is odd then $r=(n+1)/2$ and $c=(n-1)/2$. If $n$ is even, then $r=(n+2)/2$ and $c=(n-2)/2$.