  We compute the effect of primordial non-Gaussianity on the halo mass
function, using excursion set theory. In the presence of non-Gaussianity the
stochastic evolution of the smoothed density field, as a function of the
smoothing scale, is non-markovian and beside "local" terms that generalize
Press-Schechter (PS) theory, there are also "memory" terms, whose effect on the
mass function can be computed using the formalism developed in the first paper
of this series. We find that, when computing the effect of the three-point
correlator on the mass function, a PS-like approach which consists in
neglecting the cloud-in-cloud problem and in multiplying the final result by a
fudge factor close to 2, is in principle not justified. When computed correctly
in the framework of excursion set theory, in fact, the "local" contribution
vanishes (for all odd-point correlators the contribution of the image gaussian
cancels the Press-Schechter contribution rather than adding up), and the result
comes entirely from non-trivial memory terms which are absent in PS theory.
However it turns out that, in the limit of large halo masses, where the effect
of non-Gaussianity is more relevant, these memory terms give a contribution
which is the the same as that computed naively with PS theory, plus subleading
terms depending on derivatives of the three-point correlator. We finally
combine these results with the diffusive barrier model developed in the second
paper of this series, and we find that the resulting mass function reproduces
recent N-body simulations with non-Gaussian initial conditions, without the
introduction of any ad hoc parameter.
