  The odd-even staggerings (OES) on nuclear binding energies are studied
systematically within the covariant density functional (CDF) theories,
specifically the relativistic Hartree-Fock-Bogoliubov (RHFB) and the
relativistic Hartree-Bogoliubov (RHB) theories. Taking the finite-range Gogny
force D1S as an effective pairing interaction, both CDF models can provide
appropriate descriptions on the OESs of nuclear binding energies for C, O, Ca,
Ni, Zr, Sn, Ce, Gd and Pb isotopes as well as for N=50 and 82 isotones.
However, due to the inconsistence between the non-relativistic pairing
interaction and the relativistic effective Lagrangians, there exist some
systematical discrepancies from the data, i.e., the underestimated OESs in
light C and O isotopes and the overestimated ones in heavy region,
respectively. Such discrepancies can be eliminated partially by introducing a
$Z$- or $N$-dependent strength factor into the pairing force Gogny D1S. In
addition, successful descriptions of the occupation numbers of Sn isotopes are
achieved with the optimized Gogny pairing force. Furthermore, the analysis of
the systematics of both pairing effects and nuclear binding energy indicate the
requirement of an unified relativistic mechanism in both p-p and p-h channels
to improve the quantitative precision of the theory.
