6. Let the non-real complex number $z$, satisfy $z^{23}=1$. Then $\sum_{k=0}^{22} \frac{1}{1+z^{k}+z^{2 k}}=$ $\qquad$ .
6. $\frac{46}{3}$.

Notice that,
$$
\begin{array}{l}
\sum_{k=0}^{22} \frac{1}{1+z^{k}+z^{2 k}}=\frac{1}{3}+\sum_{k=1}^{22} \frac{1-z^{k}}{1-z^{3 k}} \\
=\frac{1}{3}+\sum_{k=1}^{22} \frac{1-\left(z^{24}\right)^{k}}{1-z^{3 k}}=\frac{1}{3}+\sum_{k=1}^{22} \sum_{l=0}^{7} z^{3 k l} \\
=\frac{1}{3}+(22-7)=\frac{46}{3} .
\end{array}
$$