  The static density response of the uniform electron gas is of fundamental
importance for numerous applications. Here, we employ the recently developed
\textit{ab initio} permutation blocking path integral Monte Carlo (PB-PIMC)
technique [T.~Dornheim \textit{et al.}, \textit{New J.~Phys.}~\textbf{17},
073017 (2015)] to carry out extensive simulations of the harmonically perturbed
electron gas at warm dense matter conditions. In particular, we investigate in
detail the validity of linear response theory and demonstrate that PB-PIMC
allows to obtain highly accurate results for the static density response
function and, thus, the static local field correction. A comparison with
dielectric approximations to our new \textit{ab initio} data reveals the need
for an exact treatment of correlations. Finally, we consider a superposition of
multiple perturbations and discuss the implications for the calculation of the
static response function.
