Asymptotic formula of a “hyperbolic” summation related to the Piltz divisor function

In this paper, we obtain asymptotic formula on the "hyperbolic" summation \begin{equation*} \underset{mn\leq x}{\sum }D_{k}\left( \gcd \left( m,n\right) \right) \text{ \ \ }\left( k\in \mathbb{Z}_{\geq 2}\right), \end{equation*} such that $D_{k}\left( n\right) = \dfrac{\tau _{k}\left( n\right) }{\tau_{k}^{\ast }\left( n\right) }$, where $\tau _{k}\left( n\right) =\!\!\sum\limits_{n_{1}n_{2}\ldots n_{k}=n}\!\!1$ denotes the Piltz divisor function, and $\tau _{k}^{\ast }\left( n\right) $ is the unitary analogue function of $\tau _{k}\left( n\right) $.