  We consider the zero-dispersion limit for the Benjamin-Ono equation on the
torus for bell shaped initial data. Using the approximation by truncated
Fourier series, we transform the eigenvalue equation for the Lax operator into
a problem in the complex plane. Then, we use the steepest descent method to get
asymptotic expansions of the Lax eigenvalues. As a consequence, we determine
the weak limit of solutions as the dispersion parameter goes to zero, as long
as the initial data is an even bell shaped potential.
