  We investigate the pair production of S-wave heavy quarkonium at the LHC in
the color-singlet mechanism (CSM) and estimate the contribution from the gluon
fragmentation process in the color-octet mechanism (COM) for comparison. With
the matrix elements extracted previously in the leading order calculations, the
numerical results show that the production rates are quite large for the pair
production processes at the LHC. The $p_t$ distribution of double $J/\psi$
production in the CSM is dominant over that in the COM when $p_t$ is smaller
than about 8GeV. For the production of double $\Upsilon$, the contribution of
the COM is always larger than that in the CSM. The large differences in the
theoretical predictions between the CSM and COM for the $p_t$ distributions in
the large $p_t$ region are useful in clarifying the effects of COM on the
quarkonium production. We also investigate the pair production of S-wave $B_c$
and $B_c^*$ mesons, and the measurement of these processes is useful to test
the CSM and extract the LDMEs for the $B_c$ and $B_c^*$ mesons.
