Note on fractional sum of the divisor function

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. We prove that \begin{align*} \sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+O_{\varepsilon,c}(x^{θ_c+\varepsilon}), \end{align*} where $d_c$ is a suitable constant, \begi