  We derive exact formulas for the expectation value of local observables in a
one-dimensional gas of bosons with point-wise repulsive interactions
(Lieb-Liniger model). Starting from a recently conjectured expression for the
expectation value of vertex operators in the sinh-Gordon field theory, we
derive explicit analytic expressions for the one-point $K$-body correlation
functions $\langle (\Psi^\dagger)^K(\Psi)^K\rangle$ in the Lieb-Liniger gas,
for arbitrary integer $K$. These are valid for all excited states in the
thermodynamic limit, including thermal states, generalized Gibbs ensembles and
non-equilibrium steady states arising in transport settings. Our formulas
display several physically interesting applications: most prominently, they
allow us to compute the full counting statistics for the particle-number
fluctuations in a short interval. Furthermore, combining our findings with the
recently introduced generalized hydrodynamics, we are able to study multi-point
correlation functions at the Eulerian scale in non-homogeneous settings. Our
results complement previous studies in the literature and provide a full
solution to the problem of computing one-point functions in the Lieb Liniger
model.
