by Affleck for obtaining exact asymptotic correlation functions of single-spin operators in the Heisenberg antiferromagnetic chain [@Affleck98] by combining renormalization group (RG) improved perturbation theory with the exact asymptotic results of Lukyanov and Zamolodchikov [@LZ] conjectured for $-1<\lambda<1$. However, in the case of the correlation functions of the bilinear spins, there are various operators of effective field theory that contribute equally at the $SU(2)$ antiferromagnetic point, though the exact asymptotic expression is known for $-1<\lambda<1$ for the correlation function involving only one of them [@LT]. We have to use additional symmetry arguments to obtain exact asymptotic expression of the complete uniform parts of the bilinear spin correlation functions at $\lambda=1$.

We also identify numerically the multiplicative logarithmic contribution of the staggered (leading) parts of the bilinear spin correlation functions, consistent with analytical prediction [@Affleck+89], and estimate the numerical prefactor.

Single-spin correlation functions
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Asymptotic expressions of the single-spin correlation functions of the XXZ spin-$\frac{1}{2}$ chain, $ G^a_r=\langle S^a_{0}S^a_r\rangle$, where $a=x,y,z$ and no summation with repeated indices is implied in this work, are known exactly in the gapless phase, including the numerical prefactors [@LZ; @Lukyanov99; @LT] $$\begin{aligned}
\label{ampl}
\! G^x_r=\!\frac{(-1)^r\!A_0^x}{r^{\eta}}-\frac{A_1^x}{r^{\eta+1/\eta}},\, G^z_r=\!\frac{(-1)^r\!A_1^z}{r^{1/\eta}}-\frac{1}{4\eta \pi^2 r^2},\end{aligned}$$ where $\eta=1-({\arccos{\lambda}})/{\pi}=2\pi R^2$ and $0 \le \eta \le 