On the generalised divisor problem

In this paper, we apply the Dirichlet convolution method to \(\begin{aligned} T_{k}(x)=\sum _{n \le x} d_{k}(n), \end{aligned}\) $$\begin{array}{c}\hfill T_k(x)=\sum _{n\le x}{d}_k(n),\end{array}$$ for \(k\ge 3\) $k\ge 3$ , where \(d_{k}(n)\) $d_k(n)$ is the number of ways to represent n as a product of k positive integer factors. We prove that for \(k=3\) $k=3$ , the error term \(|\Delta _3(x)| <2.968x^{2/3}\log ^{1/3}x\) $|{\mathrm{\Delta }}_3(x)|<2.968x^{2/3}{log}^{1/3}x$ for all \(x\ge 2\) $x\ge 2$ . This improves the best-known explicit result established by Bordellès for all \(x\ge 2\) $x\ge 2$ . We extend this for all \(k>3\) $k>3$ and obtain an explicit error term of the form \(\Delta _{k}(x)=O\left( x^{\frac{k-1}{k}}(\log x)^{\frac{(k-1)(k-2)}{2k}}\right) \) ${\mathrm{\Delta }}_k(x)=O\left(x^{\frac{k-1}{k}},{(logx)}^{\frac{(k-1)(k-2)}{2k}}\right)$ .