We study the polarized Bjorken sum rule at low momentum transfers in the range $0.22<Q<1.73 {rm GeV}$ with the four-loop N$^3$LO expression for the coefficient function $C_{rm Bj}(alpha_s)$ in the framework of the common QCD perturbation theory (PT) and the singularity-free analytic perturbation theory (APT). The analysis of the PT series for $C_{rm Bj}(alpha_s)$ gives a hint to its asymptotic nature manifesting itself in the region $Q<1$ GeV. It relates to the observation that the accuracy of both the three- and four-loop PT predictions happens to be at the same 10% level. On the other hand, the usage of the two-loop APT allows one to describe the precise low energy JLab data down to $Qsim 300$ MeV and gives a possibility for reliable extraction of the higher twist (HT) corrections. At the same time, above $Qsim 700$ MeV the APT two-loop order with HT is equivalent to the four-loop PT with HT compatible to zero and is adequate to current accuracy of the data.
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