We consider an ionic fluid made with two species of mobile particles carrying
either a positive or a negative charge. We derive a sum rule for the fourth
moment of equilibrium charge correlations. Our method relies on the study of
the system response to the potential created by a weak external charge
distribution with slow spatial variations. The induced particle densities, and
the resulting induced charge density, are then computed within density
functional theory, where the free energy is expanded in powers of the density
gradients. The comparison with the predictions of linear response theory
provides a thermodynamical expression for the fourth moment of charge
correlations, which involves the isothermal compressibility as well as suitably
defined partial compressibilities. The familiar Stillinger-Lovett condition is
also recovered as a by-product of our method, suggesting that the fourth moment
sum rule should hold in any conducting phase. This is explicitly checked in the
low density regime, within the Abe-Meeron diagrammatical expansions. Beyond its
own interest, the fourth-moment sum rule should be useful for both analyzing
and understanding recently observed behaviours near the ionic critical point
