Note on a sum involving the divisor function

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. In this paper, we prove that $$\sum _{n\le x^{1\mathrm{/}c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_c{x}^{1\mathrm{/}c}+{\mathcal{O}}_{\epsilon ,c}\left(x^{\max \{(2c+2)\mathrm{/}(2c^2+5c+2),5\mathrm{/}(5c+6)\}+\epsilon }\right),$$ where $d_c={\sum }_{k\ge 1}d(k)(\frac{1}{k^{1\mathrm{/}c}}-\frac{1}{(k+1{)}^{1\mathrm{/}c}})$ is a constant. This result constitutes an improvement upon that of Feng.