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Joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives

Shengbo Zhao

Abstract

In this paper, we establish joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives using the resonance method. Our results extend previous work of Aistleitner et al. (2019) and Yang (2023).

Joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives

Abstract

In this paper, we establish joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives using the resonance method. Our results extend previous work of Aistleitner et al. (2019) and Yang (2023).

Paper Structure

This paper contains 6 sections, 13 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Auxiliary Lemmas
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['thm2']}
  5. Proof of Theorem \ref{['thm3']}
  6. Proof of Theorem \ref{['thm4']}

Key Result

Theorem 1.1

Let $\ell \ge 1$ be a fixed integer. For all sufficiently large primes $q$, there exists a Dirichlet character $\chi \, (\operatorname{mod} q)$ with $\operatorname{ord}(\chi) > \ell$ such that where $C(\ell) = (\ell+1)/2 + \log_2 4$. The implied constant in the $O(\cdot)$ only depends on $\ell$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (28)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.3
  • Remark 1.5
  • Remark 1.6
  • Corollary 1.1
  • ...and 18 more