On the Generalised Divisor Problem
Sebastian Tudzi
TL;DR
This work addresses the generalized Dirichlet divisor problem by applying the Dirichlet convolution framework and the hyperbola method to analyze the sums $T_k(x)=\sum_{n\le x} d_k(n)$. It proves an explicit bound for the cubic case: $|\Delta_3(x)|<2.968\,x^{2/3}\log^{1/3}x$ for all $x\ge 1.1\times 10^{10}$, and extends the method to general $k\ge 3$ to obtain $\Delta_k(x)=O\left(x^{\frac{k-1}{k}}(\log x)^{\frac{(k-1)(k-2)}{2k}}\right)$ with explicit constants, via a constructive optimization of a splitting parameter $U$ and a sequence of lemmas controlling error terms. The results supply fully explicit main terms and error bounds that improve prior explicit estimates and have potential applications in related arithmetic problems, including bounds for class numbers and explicit estimations of divisor sums. The paper also discusses limitations of the constants and outlines future directions to push the exponent of $x$ further using analytic techniques.
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 $|Δ_3(x)|< 2.968x^{2/3}\log^{1/3}x$ for all $x\ge 2$. This improves the best-known explicit result established by Bordell{è}s for all $x\ge 2$. We extend this for all $k>3$ and obtain an explicit error term of the form $Δ_{k}(x)=O\left(x^{\frac{k-1}{k}}(\log x)^{\frac{(k-1)(k-2)}{2k}}\right)$.