Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A dichotomy for extreme values of zeta and Dirichlet L-functions

Andriy Bondarenko, Pranendu Darbar, Markus Valås Hagen, Winston Heap, Kristian Seip

Abstract

We exhibit large values of the Dedekind zeta function of a cyclotomic field on the critical line. This implies a dichotomy whereby one either has improved lower bounds for the maximum of the Riemann zeta function, or large values of Dirichlet $L$-functions on the level of the Bondarenko--Seip bound.

A dichotomy for extreme values of zeta and Dirichlet L-functions

Abstract

We exhibit large values of the Dedekind zeta function of a cyclotomic field on the critical line. This implies a dichotomy whereby one either has improved lower bounds for the maximum of the Riemann zeta function, or large values of Dirichlet -functions on the level of the Bondarenko--Seip bound.
Paper Structure (3 sections, 7 theorems, 64 equations)

This paper contains 3 sections, 7 theorems, 64 equations.

Table of Contents

  1. introduction
  2. Extremal GCD-type sums
  3. Proof of Theorem \ref{['main theorem']}

Key Result

Theorem 1

Let $\mathbb{K}=\mathbb{Q}(\omega_q)$ and $A$ be an arbitrary positive number. If $T$ is sufficiently large, then uniformly for $q\ll (\log_2 T)^A$,

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 4 more