Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Effective Brauer-Siegel theorems for Artin $L$-functions

Peter J. Cho, Robert J. Lemke Oliver, Asif Zaman

TL;DR

This work extends Stark’s effective Brauer–Siegel framework from Dedekind zeta functions to all Artin $L$-functions of a Galois extension $K/k$, proving unconditional upper and lower bounds for the leading coefficient $\kappa(\chi)$ in the Laurent expansion at $s=1$ in terms of invariants like $\chi(1)$, the analytic conductor $q(\chi)$, and the discriminant $D_K$, while accounting for exceptional zeros via the set $\Psi_{K/k}(G)$. The authors implement a short Euler-product approach, combined with a uniform Chebotarev density theorem of Thorner–Zaman, to approximate $\kappa(\chi)$ by truncated Euler products and a correction factor $\widetilde{\eta}(\chi,T)$, handling non-holomorphicity and the exceptional-zero phenomena unconditionally. They derive two main results: bounds in terms of $D_K$ that depend on possible exceptional characters, and bounds in terms of the conductor $q(\chi)$ with respect to potentially exceptional quadratic characters; together with corollaries for nonexceptional and irreducible characters and a conditional GRH-bound that mirrors the unconditional results with $\log$ replaced by $\log\log$. The work also provides illustrative Dirichlet and Dedekind cases, analyzes base-field and decomposition choices, and discusses the limitations and sharpness of these bounds in light of current knowledge on Artin $L$-functions.

Abstract

Given a number field $K \neq \mathbb{Q}$, in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function $ζ_K(s)$ and used this to give effective upper and lower bounds on the residue of $ζ_K(s)$ at $s=1$. We extend Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin $L$-functions at $s=1$ that are, up to the value of implied constants, as strong as could reasonably be expected given current progress toward the generalized Riemann hypothesis. Our bounds are completely unconditional, and rely on no unproven hypotheses about Artin $L$-functions.

Effective Brauer-Siegel theorems for Artin $L$-functions

TL;DR

This work extends Stark’s effective Brauer–Siegel framework from Dedekind zeta functions to all Artin -functions of a Galois extension , proving unconditional upper and lower bounds for the leading coefficient in the Laurent expansion at in terms of invariants like , the analytic conductor , and the discriminant , while accounting for exceptional zeros via the set . The authors implement a short Euler-product approach, combined with a uniform Chebotarev density theorem of Thorner–Zaman, to approximate by truncated Euler products and a correction factor , handling non-holomorphicity and the exceptional-zero phenomena unconditionally. They derive two main results: bounds in terms of that depend on possible exceptional characters, and bounds in terms of the conductor with respect to potentially exceptional quadratic characters; together with corollaries for nonexceptional and irreducible characters and a conditional GRH-bound that mirrors the unconditional results with replaced by . The work also provides illustrative Dirichlet and Dedekind cases, analyzes base-field and decomposition choices, and discusses the limitations and sharpness of these bounds in light of current knowledge on Artin -functions.

Abstract

Given a number field , in a now classic work, Stark pinpointed the possible source of a so-called Landau-Siegel zero of the Dedekind zeta function and used this to give effective upper and lower bounds on the residue of at . We extend Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin -functions at that are, up to the value of implied constants, as strong as could reasonably be expected given current progress toward the generalized Riemann hypothesis. Our bounds are completely unconditional, and rely on no unproven hypotheses about Artin -functions.

Paper Structure

This paper contains 14 sections, 19 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Some examples
  3. Dirichlet $L$-functions
  4. Dedekind zeta functions
  5. Choice of base field
  6. Choice of decomposition
  7. Conditional bounds
  8. Method overview
  9. Preliminaries on Artin $L$-functions
  10. Exceptional zeros and potentially exceptional characters
  11. Short Euler product approximations and the proof of Theorem 3.1
  12. Three key propositions
  13. Proofs of Theorems 1.1 and 1.2 and Corollaries 1.3 and 1.4
  14. Proof of Proposition 2.1

Key Result

Theorem 1.1

Let $K/k$ be a Galois extension of number fields with Galois group $G$, and let $\psi_{K/k}$ be the exceptional character, if it exists. For any character $\chi$ of $G$, we have that and where $\nu(\chi)=\langle \chi, \psi_{K/k} \rangle$ if $\psi_{K/k}$ exists and $\nu(\chi) = 0$ otherwise.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Theorem 3.1
  • Proposition 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 33 more