On the sum of $\Delta_{k}(n)$ in the Piltz divisor problem for $k=3$ and $k=4$

Abstract

Let $\Delta_{k}(x)$ be the error term in the classical asymptotic formula for the sum $\sum_{n\leq x}d_{k}(n)$, where $d_{k}(n)$ is the number of ways $n$ can be written as a product of $k$ factors. We study the analytic properties of the Dirichlet series $\sum_{n=1}^{\infty}\Delta_{k}(n)n^{-s}$ and use Perron's formula to estimate the sums $\sum_{n\leq x}\Delta_{3}(n)$ and $\sum_{n\leq x}\Delta_{4}(n)$ for large $x>0$.