Sums of Apostol's Möbius functions of order $k$

In 1970, T. M. Apostol introduced the Möbius function ${\mu }_k$ of order $k$ for all positive integer $k$, as a generalization of the Möbius function $\mu ={\mu }_1$. For any integer $k\ge 2$, he proved ${\sum }_{n\le x}{\mu }_k(n)=A_{k}x+O_k(x^{1\mathrm{/}k}\log x)$ where $A_k$ is a positive constant. In 2001, A. Bege conjectured both the conditional and unconditional estimates for the sum ${\sum }_{n\le x,(n,q)=1}{\mu }_k(n)$ for any positive integer $q$. In this paper, we give affirmative solutions to the conditional version of Bege’s conjecture completely and the unconditional one partially. We also give a mean square estimate for the error term.