  We study the density-density correlation function of the 1D Lieb-Liniger
model and obtain an exact expression for the small momentum limit of the static
correlator in the thermodynamic limit. We achieve this by summing exactly over
the relevant form factors of the density operator in the small momentum limit.
The result is valid for any eigenstate, including thermal and non-thermal
states. We also show that the small momentum limit of the dynamic structure
factors obeys a generalized detailed balance relation valid for any equilibrium
state.
