  Kummer's conjecture predicts the asymptotic growth of the relative class
number of prime cyclotomic fields. We substantially improve the known bounds of
Kummer's ratio under three scenarios: no Siegel zero, presence of Siegel zero
and assuming the Riemann Hypothesis for the Dirichlet $L$-series attached to
odd characters only. The numerical work in this paper extends and improves on
our earlier preprint (arXiv:1908.01152) and demonstrates our theoretical
results.
