  We derive distance-dependent estimators for two-center and three-center
electron repulsion integrals over a short-range Coulomb potential,
$\textrm{erfc}(\omega r_{12})/r_{12}$. These estimators are much tighter than
one based on the Schwarz inequality and can be viewed as a complement to the
distance-dependent estimators for four-center short-range Coulomb integrals and
for two-center and three-center full Coulomb integrals previously reported.
Because the short-range Coulomb potential is commonly used in solid-state
calculations, including those with the HSE functional and with our recently
introduced range-separated periodic Gaussian density fitting, we test our
estimators on a diverse set of periodic systems using a wide range of the
range-separation parameter $\omega$. These tests demonstrate the robust
tightness of our estimators, which are then used with integral screening to
calculate periodic three-center short-range Coulomb integrals with linear
scaling in system size.
