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We present new results for the jet-veto efficiency and zero-jet cross sectionin Higgs production through gluon fusion. We incorporate the N$^3$LOcorrections to the total cross section, the NNLO corrections to the 1-jet rate,NNLL resummation for the jet $p_t$ and LL resummation for the jet radiusdependence. Our results include known finite-mass corrections and are obtainedusing the jet-veto efficiency method, updated relative to earlier work to takeinto account what has been learnt from the new precision calculations that weinclude. For 13 TeV collisions and using our default choice for therenormalisation and factorisation scales, $\mu_0=m_H/2$, the matched predictionfor the jet-veto efficiency increases the pure NNNLO prediction by about 2% andthe two have comparable uncertainties. Relative to NNLO+NNLL results, the newprediction is 2% smaller and the uncertainty reduces from more than 10% to lessthan 5%. Results are also presented for the central scale $\mu_0=m_H$.

Jet-vetoed Higgs cross section in gluon fusion at N3LO+NNLL with small-R resummation