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The Brauer-Siegel ratio for prime cyclotomic fields

Neelam Kandhil, Alessandro Languasco, Pieter Moree

TL;DR

This work determines explicit Brauer-Siegel-type bounds for prime cyclotomic fields by analyzing $\mathcal{R}(q)=\prod_{\chi\neq \chi_0} L(1,\chi)$, linking it to $h(q)\operatorname{Reg}(q)$ via $H(q)$ and distinguishing the Siegel-zero scenario. The authors convert Tatuzawa-type bounds into explicit expressions involving the exponential integral $E_1(1-\beta_0)$ and develop a rigorous decomposition of $\log \mathcal{R}(q)$ into prime-sum contributions, carefully treating prime powers. They provide thorough numerical verification up to $q\le 10^7$ using FFT methods, observing that $\mathcal{R}(q)$ closely follows $c/(\log q)^{3/4}$ and illustrating the distribution with plots and histograms. Furthermore, the paper connects these sums to Meissel-Mertens constants in arithmetic progressions, drawing parallels to classical Mertens-type products and offering a framework for precise, computable bounds. These results yield asymptotic implications for $\log(h(q)\operatorname{Reg}(q))$, improving Brauer-Siegel-type predictions for prime cyclotomic fields and highlighting the interplay between zero-free regions and explicit arithmetic constants.

Abstract

The Brauer-Siegel theorem concerns the size of the product of the class number and the regulator of a number field $K$. We derive bounds for this product in case $K$ is a prime cyclotomic field, distinguishing between whether there is a Siegel zero or not. In particular, we make a result of Tatuzawa (1953) more explicit. Our theoretical advancements are complemented by numerical illustrations that are consistent with our findings.

The Brauer-Siegel ratio for prime cyclotomic fields

TL;DR

This work determines explicit Brauer-Siegel-type bounds for prime cyclotomic fields by analyzing , linking it to via and distinguishing the Siegel-zero scenario. The authors convert Tatuzawa-type bounds into explicit expressions involving the exponential integral and develop a rigorous decomposition of into prime-sum contributions, carefully treating prime powers. They provide thorough numerical verification up to using FFT methods, observing that closely follows and illustrating the distribution with plots and histograms. Furthermore, the paper connects these sums to Meissel-Mertens constants in arithmetic progressions, drawing parallels to classical Mertens-type products and offering a framework for precise, computable bounds. These results yield asymptotic implications for , improving Brauer-Siegel-type predictions for prime cyclotomic fields and highlighting the interplay between zero-free regions and explicit arithmetic constants.

Abstract

The Brauer-Siegel theorem concerns the size of the product of the class number and the regulator of a number field . We derive bounds for this product in case is a prime cyclotomic field, distinguishing between whether there is a Siegel zero or not. In particular, we make a result of Tatuzawa (1953) more explicit. Our theoretical advancements are complemented by numerical illustrations that are consistent with our findings.
Paper Structure (12 sections, 5 theorems, 82 equations, 3 figures)

This paper contains 12 sections, 5 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Siegel zeros
  5. A useful lemma
  6. Proof of Theorem \ref{['rq-direct3']}
  7. Proof of Theorem \ref{['rq-direct3']}, Part \ref{['Siegelzero3']}
  8. Proof of Theorem \ref{['rq-direct3']}, Part \ref{['nosiegelzero3']}
  9. Numerical results
  10. Comments on the plots and on the histograms
  11. Computing the prime sums over Dirichlet characters
  12. Connections and analogies with the Mertens' constants in arithmetic progressions

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 $\ell$) such that the following statements are true:

Figures (3)

  • Figure 1: The values of $\mathcal{R}(q)$, $q$ prime, $3\le q\le 10^7$. The maximal value (red dot) is attained at $q=3$ and its value is $0.604599\dots$; much larger than the other plotted values. The red dashed line represents the mean value.
  • Figure 2: The values of $\mathcal{R}(q) ( \log q)^{3/4}$, $q$ prime, $3\le q\le 10^7$. The red dashed line represents the mean value.
  • Figure 3: On the left: the values of $\mathcal{R}(q)$ (cerulean bars), $q$ prime, $3\le q\le 10^7$, but the contributions of the primes $q \ge 5$ such that $2q+1$ is prime (green bars) or $2q-1$ is prime (yellow bars) are superimposed. On the right: idem, but for the normalized values $\mathcal{R}(q) (\log q)^{3/4}$. The red dashed lines represent the mean values.

Theorems & Definitions (11)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem : Maynard
  • Theorem : Dusart
  • Lemma 1
  • proof
  • Remark 4: Precise numerical approximation of ${\mathcal{A}}$
  • Remark 5
  • ...and 1 more