We revisit the heavy quarkonium leptonic decays
and
using the Bethe-Salpeter method. The emphasis is on the relativistic corrections. For the
decays, the relativistic effects are
%,
%,
%,
% and
%, respectively. For the
decays, the relativistic effects are
%,
%,
%,
% and
%, respectively. Thus, the relativistic corrections are large and important in heavy quarkonium leptonic decays, especially for the highly excited charmonium. Our results for
are consistent with the experimental data.