  We calculate the screening function in bilayer graphene (BLG) both in the
intrinsic (undoped) and the extrinsic (doped) regime within random phase
approximation, comparing our results with the corresponding single layer
graphene (SLG) and the regular two dimensional electron gas (2DEG). We find
that the Kohn anomaly is strongly enhanced in BLG. We also discuss the Friedel
oscillation and the RKKY interaction, which are associated with the
non-analytic behavior of the screening function at $q=2k_F$. We find that the
Kohn anomaly, the Friedel oscillation, and the RKKY interaction are all
qualitatively different in the BLG compared with the SLG and the 2DEG.
