In this paper, we apply the Dirichlet convolution method to \begin{ equation*} T_{ k}(x)=\sum_{ n \leq x} d_{ k}(n), \end{ equation*} for $k\ge 3$, where $d_{ k}(n)$ is the number of ways to represent $n$ as a product of $k$ positive integer factors. We prove that for $k=3$, the error term $|\Delta_3(x)|< 2.968x^{ 2/3}\log^{ 1/3}x$ for all $x\ge 2$. This improves the best-known explicit result established by Bordell{ \`e}s for all $x\ge 2$. We extend this for all $k>3$ and obtain an explicit error term of the form $\Delta_{ k}(x)=O\left(x^{ \frac{ k-1}{ k}}(\log x)^{ \frac{ (k-1)(k-2)}{ 2k}}\right)$.
Categories: math.NT