  We review the formalism of primordial black holes (PBHs) production and show
that the mass variance at horizon crossing has been systematically
overestimated in previous studies. We derive the correct expression. The
difference is maximal at the earliest formation times and still very
significant for PBH masses $\sim 10^{15}$g, an accurate estimate requiring
numerical calculations. In particular, this would lead to weaker constraints on
the spectral index $n$. We then derive constraints on inflationary models from
the fact that primordial black holes must not overclose the Universe. This is
done both for the scale-free case of the power spectrum studied earlier and for
the case where a step in the mass variance is superimposed. In the former case
we find various constraints on $n$, depending on the parameters. In the latter
case these limits can be much more strengthened, so that one could find from an
observational limit on $n$ a constraint on the allowed height of the step.
