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 https://arxiv.org/abs/1908.01152 and demonstrates our theoretical results.
The Kummer ratio of the relative class number for prime cyclotomic fields
Alessandro Languasco;
2024
Abstract
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 https://arxiv.org/abs/1908.01152 and demonstrates our theoretical results.
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