  We present extensive numerical results for the thermodynamic density of
states (i.e. quantum capacitance) of a two-dimensional massless Dirac fermion
fluid in a doped graphene sheet. In particular, by employing the random phase
approximation, we quantify the impact of screening exerted by a metal gate
located nearby a graphene flake. Finally, we calculate the spin- and
circularly-symmetric Landau parameter, which can be experimentally extracted
from independent measurements on the same setup of the quantum capacitance and
quasiparticle velocity.
