  We consider the K-body local correlations in the (repulsive) 1D Bose gas for
general K, both at finite size and in the thermodynamic limit. Concerning the
latter we develop a multiple integral formula which applies for arbitrary
states of the system with a smooth distribution of Bethe roots, including the
ground state and finite temperature Gibbs-states. In the cases K<=4 we perform
the explicit factorization of the multiple integral. In the case of K=3 we
obtain the recent result of Kormos et.al., whereas our formula for K=4 is new.
Numerical results are presented as well.
