  The Lieb-Liniger model is a prototypical integrable model and has been turned
into the benchmark physics in theoretical and numerical investigations of low
dimensional quantum systems.
  In this note, we present various methods for calculating local and nonlocal
$M$-particle correlation functions, momentum distribution and static structure
factor. In particular, using the Bethe ansatz wave function of the strong
coupling Lieb-Liniger model, we analytically calculate two-point correlation
function, the large moment tail of momentum distribution and static structure
factor of the model in terms of the fractional statistical parameter $\alpha
=1-2/\gamma$, where $\gamma$ is the dimensionless interaction strength. We also
discuss the Tan's adiabatic relation and other universal relations for the
strongly repulsive Lieb-Liniger model in term of the fractional statistical
parameter.
