The Kummer ratio of the relative class number for prime cyclotomic fields

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 (arXiv:1908.01152) and demonstrates our theoretical results.

arXiv comment23 pages, 4 figures, 3 tables; to appear in Journal of Mathematical Analysis and Applications

References in corpus (4)

• A fast algorithm to compute the Ramanujan-Deninger gamma-function and some number-theoretic applications
• Efficient computation of the Euler-Kronecker constants of prime cyclotomic fields
• Numerical verification of Littlewood's bounds for $|L(1,\chi )|$
• A unified strategy to compute some special functions of number-theoretic interest

Cited by in corpus (1)

• Relative class numbers and Euler-Kronecker constants of maximal real cyclotomic subfields