O(1) contribution of saddle point fluctuations to the free energy of Bethe Ansatz systems

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.