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 $D_{k}\left( n\right) = \dfrac{\tau _{k}\left( }{\tau_{k}^{\ast }$, where $\tau =\!\!\sum\limits_{n_{1}n_{2}\ldots n_{k}=n}\!\!1$ denotes Piltz divisor function, and _{k}^{\ast $ is unitary analogue function of $.