I will discuss logarithmic corrections to various CFT partition functions in the context of the AdS_4/CFT_3 correspondence for theories arising on the worldvolume of M2-branes. I will use four-dimensional gauged supergravity and heat kernel methods to derive general expressions for the logarithmic corrections to the gravitational on-shell action or black hole entropy for a number of different supergravity backgrounds. I will outline several subtleties and puzzles in these calculations and will show how they provide non-trivial precision tests of the AdS/CFT correspondence. These results have important implications for the existence of scale separated AdS vacua in string theory and for effective field theory in AdS more generally.
