On the Generalised Divisor Problem

Abstract

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 $\Delta_{k}(x)=O\left(x^{\frac{k-1}{k}}(\log x)^{\frac{(k-1)(k-2)}{2k}}\right)$.