We study the screening of a macroscopic electric field in a crystalline dielectric. Density-functional perturbation theory provides the static dielectric constant (or tensor) as a bulk property; we give a formulation which extends the local-density approximation, and specifically we discuss its implementation within gradient-corrected schemes. We briefly consider the relevance (if any) of the so-called ''gap problem'' to static linear response. As a case study, we perform an ab initio calculation of the dielectric constant in silicon within a popular gradient-corrected local-density scheme. We find that the gradient corrections reduce the discrepancy found so far between local-density predictions and experiments in covalently bonded materials. The amount of this reduction is sizable if the calculations are performed at the experimental equilibrium lattice constant of the crystal, while however, it is only marginal when the calculations are carried out, at the calculated lattice constants, consistently within each given theoretical scheme.
