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

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 $x \gt 0$.