Estimates of the coefficients of mayer expansions on the boundary of the infrared region

Classical gas on the lattice and with the two-particle interaction $\Phi(x-y)$ is considered. Rejecting the regularity but assuming the stability and boundedness of the Fourier transformation $\widetilde\Phi(k)$ we prove the estimate $|b_n|<C^nn!$ for the coefficients of the Mayer expansion $\sum b_nz^n$ e. g. for the first correlation function.