A perturbation theory is developed for the logarithm of the normalization integral (or partition function) of an N-body system which is either a Bose liquid in its ground state or a classical fluid in the canonical ensemble. The perturbation in the former is an n-body factor in the ground state wavefunction and in the latter is an n-body potential. The normalization function serves as a generating function for the cumulants (or static correlation functions) of the density fluctuation operator ρk. The expansions of the perturbed partition function and correlation functions are shown to be linked cluster expansions involving the correlation functions in the unperturbed system; each term in these expansions is manifestly of the proper order in N. Several approximations involving truncations of the cumulants and/or resummation of part of the terms in the linked cluster expansions are discussed.