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.
Categories: math.NT