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

Neelam Kandhil, Alessandro Languasco, Pieter Moree, Sumaia Saad Eddin, Alisa Sedunova

TL;DR

This work analyzes Kummer's conjecture on the relative class number of prime cyclotomic fields by studying the Kummer ratio $R(q)=h_1(q)/G(q)$, which equals $\prod_{\chi(-1)=-1}L(1,\chi)$. The authors derive a precise connection between $R(q)$ and prime-power sums, prove a key lemma bounding auxiliary sums, and establish explicit bounds under three regimes: no Siegel zero, existence of a Siegel zero, and RH for odd Dirichlet L-series. They provide a detailed proof of the main bound (Theorem rq-direct), quantify the effects of Siegel zeros via $E_1(1-\beta_0)$, and show that under RH$_{\text{odd}}$ the bound improves to $\max\{R(q),R(q)^{-1}\} \le e^{0.41}\log q$. Complementing the theory, they develop an efficient FFT-based algorithm to compute $r(q)=\log R(q)$ and produce extensive numerical data, including distributions and maximal/minimal champions, thereby strengthening the empirical understanding of Kummer’s ratio.

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.

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

TL;DR

This work analyzes Kummer's conjecture on the relative class number of prime cyclotomic fields by studying the Kummer ratio , which equals . The authors derive a precise connection between and prime-power sums, prove a key lemma bounding auxiliary sums, and establish explicit bounds under three regimes: no Siegel zero, existence of a Siegel zero, and RH for odd Dirichlet L-series. They provide a detailed proof of the main bound (Theorem rq-direct), quantify the effects of Siegel zeros via , and show that under RH the bound improves to . Complementing the theory, they develop an efficient FFT-based algorithm to compute and produce extensive numerical data, including distributions and maximal/minimal champions, thereby strengthening the empirical understanding of Kummer’s ratio.

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 -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.
Paper Structure (15 sections, 6 theorems, 92 equations, 4 figures, 3 tables)

This paper contains 15 sections, 6 theorems, 92 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Kummer's conjecture
  3. Preliminaries
  4. Prime number distribution
  5. Siegel zeros
  6. Admissible sets of large measure
  7. Connecting Kummer's ratio conjecture with prime power sums
  8. A useful lemma
  9. Proof of Theorem \ref{['rq-direct']}
  10. Remarks on Theorem \ref{['rq-direct']}
  11. An efficient algorithm to compute Rq
  12. Computations with trivial summing over a (slower, more digits available).
  13. FFT-Decimation in frequency
  14. FFT accuracy estimate
  15. Comments on the plots and on the histograms

Key Result

Theorem 1

Let $\ell(q)$ be a function that tends arbitrarily slow and monotonically to infinity as $q$ tends to infinity. There is an effectively computable prime $q_0$ (possibly depending on $\ell$) and an effectively computable prime $q_1$ such that the following statements are true:

Figures (4)

  • Figure 1: The values of $R(q)$, $q$ prime, $3\leqslant q\leqslant 10^7$. The red dashed line represents the mean value.
  • Figure 2: On the left: the values of $r(q)$;
  • Figure 3: On the left: the same histograms of Figure \ref{['fig2']} but the contributions of the primes $q$ such that $2q+1$ is prime or $2q-1$ is prime (the "spikes") are superimposed. On the right: the contributions of the primes $q$ such that $4q+1$ is prime or $4q-1$ is prime (the "spikes") are superimposed.
  • Figure 4: On the left: the histogram for $r(q)$, $q$ prime, $5\leqslant q\leqslant 10^7$, such that $2q\pm1$ are composite; on the right: the same with both $2q\pm1$ and $4q\pm1$ that are composite numbers. The red dashed lines represent the mean values.

Theorems & Definitions (24)

  • Definition 1
  • Conjecture 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Conjecture 2: Elliott-Halberstam
  • Conjecture 3: RH$_{\textrm{odd}}(q)$
  • Conjecture 4: Hardy-Littlewood HardyL1923, lower bound version
  • Conjecture 5: Hardy-Littlewood HardyL1923 for Sophie Germain primes
  • ...and 14 more