Explicit and Effective Estimates for the error term in the Generalised Divisor Problem

TL;DR

This paper provides explicit and effective estimates for the error term in the Generalised Divisor Problem for all integers k ≥ 2, improving known bounds and offering a method for deriving bounds for all k ≥ 3.

Contribution

It introduces explicit bounds for the error term Δ_k(x) for all integers k ≥ 2, with improved powers and a general method for all k ≥ 3.

Findings

Improved explicit bounds for Δ_k(x) for k in [3,9]

Effective bounds applicable to all integers k ≥ 3

Enhanced understanding of the error term in the Generalised Divisor Problem

Abstract

In this article, we obtain effective estimates for the error term $\Delta_{k}(x)$ for all integers $k \geq2$, and completely explicit estimates for integers $k \in [3,9]$. The explicit results improve the powers of $x$ appearing in the known explicit bounds for $\Delta_{k}(x)$, and the effective bounds provide a method to derive such bounds for all integers $k\geq 3$.