Explicit and Effective Estimates for 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 $k\geq 3$.