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

Let ${\mathrm{\Delta }}_k(x)$ be the error term in the classical asymptotic formula for the sum ${\sum }_{n\le x}{d}_k(n)$, 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 ${\sum }_{n=1}^{\mathrm{\infty }}{\mathrm{\Delta }}_k(n)n^{-s}$ and use Perron’s formula to estimate the sums ${\sum }_{n\le x}{\mathrm{\Delta }}_3(n)$ and ${\sum }_{n\le x}{\mathrm{\Delta }}_4(n)$ for large $x>0$.