  We study the fluctuations of the total current for the partially asymmetric
exclusion process in the scaling of a weak asymmetry (asymmetry of order the
inverse of the size of the system) using Bethe Ansatz. Starting from the
functional formulation of the Bethe equations, we obtain for all the cumulants
of the current both the leading and next-to-leading contribution in the size of
the system.
