  Gluon-induced processes such as Higgs production typically exhibit large
perturbative corrections. These partially arise from large virtual corrections
to the gluon form factor, which at timelike momentum transfer contains Sudakov
logarithms evaluated at negative arguments $\ln^2(-1) = -\pi^2$. It has been
observed that resumming these terms in the timelike form factor leads to a much
improved perturbative convergence for the total cross section. We discuss how
to consistently incorporate the resummed form factor into the perturbative
predictions for generic cross sections differential in the Born kinematics,
including in particular the Higgs rapidity spectrum. We verify that this indeed
improves the perturbative convergence, leading to smaller and more reliable
perturbative uncertainties, and that this is not affected by cancellations
between resummed and unresummed contributions. Combining both fixed-order and
resummation uncertainties, the perturbative uncertainty for the total cross
section at N$^3$LO$+$N$^3$LL$^\prime_\varphi$ is about a factor of two smaller
than at N$^3$LO. The perturbative uncertainty of the rapidity spectrum at
NNLO$+$NNLL$^\prime_\varphi$ is similarly reduced compared to NNLO. We also
study the analogous resummation for quark-induced processes, namely Higgs
production through bottom quark annihilation and the Drell-Yan rapidity
spectrum. For the former the resummation leads to a small improvement, while
for the latter it confirms the already small uncertainties of the fixed-order
predictions.
