We derive distance-dependent estimators for two-center and three-center electron repulsion integrals over a short-range Coulomb potential, erfc(ωr12)/r12. These estimators are much tighter than the ones 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 Heyd-Scuseria-Ernzerhof 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 ω. 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.