  In this paper, we present an improved analysis of the top-quark pair
production via the process $e^{+}e^{-}\to \gamma^{*}\to t\bar{t}$ near the
threshold region up to next-to-next-to-next-to-leading order (N$^3$LO) QCD
corrections. Near the threshold region, the top-quark velocity $v$ tends to
zero, leading to Coulomb singularity. To achieve a reasonable prediction in the
threshold region, we reconstruct the analytical expression for the
Coulomb-terms up to N$^{3}$LO accuracy by using the PSLQ algorithm, whose
numerical values agree well with the previous N$^3$LO-level calculations. It is
found that the N$^{3}$LO series still has sizable renormalization scale
dependence, and to improve the precision of the series, we apply the Principle
of Maximum Conformality to eliminate such scale dependence. After that, the
Coulomb part is resummed into a Sommerfeld-Gamow-Sakharov factor, which finally
leads to a much more reasonable behavior near the threshold region.
