On the Summatory Function of $d_3(n)$

In this article, we refine the method of our earlier work with N. Paloj{\"a}rvi to obtain a sharper explicit bound for the error term $\Delta_{3}(x)$ associated with the summatory function of $d_{3}(n)$. We prove that \begin{equation*} |\Delta_3(x)| < \begin{cases} 0.6901\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & 3.682\cdot 10^{31}\le x < 4.133\cdot 10^{87},\\[4pt] 0.2067\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & 4.133\cdot 10^{87} \le x < 1.597\cdot 10^{98},\\[4pt] 0.1947\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & x \ge 1.597\cdot 10^{98}. \end{cases} \end{equation*} These explicit results improve the exponent of $x$ from $2/3$, due to Tudzi, and $859/1400$, due to Paloj{\"a}rvi and Tudzi, to $1/2$, giving the best known bound for all $x\ge 3.682\cdot 10^{31}$.