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Explicit conditional bounds for the residue of a Dedekind zeta-function at $s=1$

Stephan Ramon Garcia, Loïc Grenié, Ethan Simpson Lee, Giuseppe Molteni

Abstract

We prove new explicit conditional bounds for the residue at $s=1$ of the Dedekind zeta-function associated to a number field. Our bounds are concrete and all constants are presented with explicit numerical values.

Explicit conditional bounds for the residue of a Dedekind zeta-function at $s=1$

Abstract

We prove new explicit conditional bounds for the residue at of the Dedekind zeta-function associated to a number field. Our bounds are concrete and all constants are presented with explicit numerical values.

Paper Structure

This paper contains 16 sections, 9 theorems, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Explicit bounds for Psi(x)
  4. Explicit bounds for functions over primes
  5. Artin L-functions
  6. Key identities
  7. Explicit short sum relationship
  8. Step I
  9. Step II
  10. Step III
  11. Step IV
  12. Step V
  13. Proof of Theorem \ref{['Theorem:Main']}
  14. Setup
  15. Upper bound
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that $\mathbb{K}$ is a number field such that $|\Delta_{\mathbb{K}}|\geq 14$, the GRH for $\zeta_{\mathbb{K}}$ is true, and $\zeta_{\mathbb{K}}/\zeta$ is entire. Then

Figures (2)

  • Figure 1: Proof of Lemma \ref{['Lemma:Handy']}.
  • Figure 2: The contours $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, $\Gamma_4$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 2.1
  • proof
  • Lemma 2.2: L.--Nosal LeeNosal
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 11 more