Searching for a C-function on the three-dimensional sphere

We present a detailed analytic study on the three-dimensional sphere of the most popular candidates for C-functions, both for Dirac and scalar free massive fields. We discuss to which extent the effective action, the Renyi entanglement entropy and the renormalized entanglement entropy fulfill the conditions expected from C-functions. In view of the absence of a good candidate in the case of the scalar field, we introduce a new candidate, which we call the modified effective action, and analyze its pros and cons.