  We develop a method to calculate the contribution of the saddle-point
fluctuations to the partition function of systems soluble by the Bethe Ansatz.
Using this method we give the O(1) corrections to the free energy of the 1D
repulsive delta Bose gas both for periodic boundary conditions and for the open
end case. We also generalize our method to more complicated systems and discuss
the case of XXZ Heisenberg chain in more details.
