  We derive the three loop soft function for inclusive Higgs production in
gluon fusion, and use it to perform the resummation of the Higgs cross section
at N$^3$LL$^\prime$ in SCET. We improve the accuracy of the resummation by
including contributions of collinear origin. We include finite top, bottom and
charm mass effect where available. These results are available through the
public code ResHiggs.
