On Mobius and Liouville functions of order $k$

Let $F$ be a number field, $k$ a positive integer. In this paper, we define the Mobius and Liouville functions of order $k$ in $F$. We give a formula about the partial sums of them by using elementary number theory and complex analysis. Moreover, we also consider the number of $k$-free ideals of the integer ring of $F$.