  We derive three exact sum rules for the spectral function of the
electromagnetic current with zero spatial momentum at finite temperature. Two
of them are derived in this paper for the first time. We explicitly check that
these sum rules are satisfied in the weak coupling regime and examine which sum
rule is sensitive to the transport peak in the spectral function at low energy
or the continuum at high energy. Possible applications of the three sum rules
to lattice computations of the spectral function and transport coefficients are
also discussed: We propose an ansatz for the spectral function that can be
applied to all three sum rules and fit it to available lattice data of the
Euclidean vector correlator above the critical temperature. As a result, we
obtain estimates for both the electrical conductivity $\sigma$ and the second
order transport coefficient $\tau_J$.
