  We present an efficient implementation of periodic Gaussian density fitting
(GDF) using the Coulomb metric. The three-center integrals are divided into two
parts by range-separating the Coulomb kernel, with the short-range part
evaluated in real space and the long-range part in reciprocal space. With a few
algorithmic optimizations, we show that this new method -- which we call
range-separated GDF (RSGDF) -- scales sublinearly to linearly with the number
of $k$-points for small to medium-sized $k$-point meshes that are commonly used
in periodic calculations with electron correlation. Numerical results on a few
three-dimensional solids show about $10$-fold speedups over the previously
developed GDF with little precision loss. The error introduced by RSGDF is
about $10^{-5}~E_{\textrm{h}}$ in the converged Hartree-Fock energy with
default auxiliary basis sets and can be systematically reduced by increasing
the size of the auxiliary basis with little extra work.
  [The article has been accepted by The Journal of Chemical Physics.]
