On the generalised divisor problem
Abstract
Abstract In this paper, we apply the Dirichlet convolution method to $$\begin{aligned} T_{k}(x)=\sum _{n \le x} d_{k}(n), \end{aligned}$$ T k ( x ) = ∑ n ≤ x d k ( n ) , for $$k\ge 3$$ k ≥ 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 $$|\Delta _3(x)| 3$$ k > 3 and obtain an explicit error term of the form $$\Delta _{k}(x)=O\left( x^{\frac{k-1}{k}}(\log x)^{\frac{(k-1)(k-2)}{2k}}\right) $$ Δ k ( x ) = O x k - 1
Fields of Study & Subjects
Analytic Number Theory Research
Advanced Harmonic Analysis Research
Limits and Structures in Graph Theory
Publication details
- Journal / source
- The Ramanujan Journal
- Volume
- 70
- Issue
- 3
- Publication date
- 12 June 2026
- Language
- EN