  Precise knowledge of the static density response function (SDRF) of the
uniform electron gas (UEG) serves as key input for numerous applications, most
importantly for density functional theory beyond generalized gradient
approximations. Here we extend the configuration path integral Monte Carlo
(CPIMC) formalism that was previously applied to the spatially uniform electron
gas to the case of an inhomogeneous electron gas by adding a spatially periodic
external potential. This procedure has recently been successfully used in
permutation blocking path integral Monte Carlo simulations (PB-PIMC) of the
warm dense electron gas [Dornheim \textit{et al.}, Phys. Rev. E in press,
arXiv:1706.00315], but this method is restricted to low and moderate densities.
Implementing this procedure into CPIMC allows us to obtain exact finite
temperature results for the SDRF of the electron gas at \textit{high to
moderate densities} closing the gap left open by the PB-PIMC data. In this
paper we demonstrate how the CPIMC formalism can be efficiently extended to the
spatially inhomogeneous electron gas and present the first data points.
Finally, we discuss finite size errors involved in the quantum Monte Carlo
results for the SDRF in detail and present a solution how to remove them that
is based on a generalization of ground state techniques.
