  We investigate the odd-even staggering (OES) in reaction cross sections of
weakly bound nuclei with a Glauber theory, taking into account the pairing
correlation with the Hartree-Fock-Bogoliubov (HFB) method. We first discuss the
pairing gap in extremely weakly bound nuclei and show that the pairing gap
persists in the limit of zero separation energy limit even for single-particle
orbits with the orbital angular momenta $l=0$ and $l=1$. We then introduce the
OES parameter defined as the second derivative of reaction cross sections with
respect to the mass number, and clarify the relation between the magnitude of
OES and the neutron separation energy. We find that the OES parameter increases
considerably in the zero separation energy limit for $l=0$ and $l=1$
single-particle states, while no increase is found for higher angular momentum
orbits with {\it e.g.,} $l=3$. We point out that the increase of OES parameter
is also seen in the experimental reaction cross sections for Ne isotopes, which
is well accounted for by our calculations.
