  Exact asymptotic expressions of the uniform parts of the two-point
correlation functions of bilinear spin operators in the Heisenberg
antiferromagnetic spin-$\frac{1}{2}$ chain are obtained. Apart from the
algebraic decay, the logarithmic contribution is identified, and the numerical
prefactor is determined. We also confirm numerically the multiplicative
logarithmic correction of the staggered part of the bilinear spin operators
$\langle\langle S^{a}_0S^{a}_{1}S^{b}_{r}S^{b}_{r+1} \rangle\rangle=(-1)^rd/(r
\ln^{\frac{3}{2}}r) +(3\delta_{a,b}-1) \ln^2r /(12 \pi^4 r^4)$, and estimate
the numerical prefactor as $d\simeq 0.067$. The relevance of our results for
ground state fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless
quantum phase transition points in one-dimensional systems is discussed at the
end of our work.
