A generalization of Apostol's Möbius functions of order \(k\) (Q2754931)

The author introduces the following generalization of Apostol's Möbius functions \(\mu_k(n)\) of order \(k\). Let \(m\) and \(k\) be integers with \(m\geq k\geq 1\). If \(m=k\) then \(\mu_{k,m}(n)=\mu_k(n)\) else NEWLINE\[NEWLINE \mu_{k,m}(n)=\begin{cases} 1 & \text{if} \;n=1\\ 1 & \text{if} \;p^k|n\;\text{for each prime}\\ (-1)^r & \text{if} \;n=p^m_1\dots p^m_r \prod_{i>r}p^{\alpha_i}_i\;\text{with} 0\leq \alpha_i<k\\ 0 & \text{otherwise}.\end{cases} NEWLINE\]NEWLINE He gives two asymptotic formulas (which are uniform in \(x\), \(n\) and \(k\)) for the partial sums \(\sum_{r\leq x,(r,n)=1}\mu_{k,m}(n)\). This article has close connections with \textit{D. Suryanarayana} [Pac. J. Math. 68, 277-281 (1977; Zbl 0349.10037)].