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Large values of logarithmic derivatives of quadratic Dirichlet $L$-functions

Zikang Dong, Haidong Li

Abstract

In this article, we apply the resonance method to derive conditional Omega results for logarithmic derivatives of quadratic Dirichlet $L$-functions. We improve a previous result of Mortada and Murty \cite{MM13}, as well as generalize some results of Yang \cite{yang2023omegatheoremslogarithmicderivatives}.

Large values of logarithmic derivatives of quadratic Dirichlet $L$-functions

Abstract

In this article, we apply the resonance method to derive conditional Omega results for logarithmic derivatives of quadratic Dirichlet -functions. We improve a previous result of Mortada and Murty \cite{MM13}, as well as generalize some results of Yang \cite{yang2023omegatheoremslogarithmicderivatives}.
Paper Structure (6 sections, 7 theorems, 75 equations)

This paper contains 6 sections, 7 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminary Lemmas
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['theorem:measurelargeofderoflogL']}
  5. Proof of Theorem of \ref{['thm3']}
  6. Proof of Theorem \ref{['thm4']}

Key Result

Theorem 1

For sufficiently large $N$, under GRH, we have for the constant where $0<\delta<\frac{1}{4}$ is any fixed small number, and $\gamma$ is the Euler constant.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • Remark 2
  • Lemma 1
  • Lemma 2: 0Large Lemma 1
  • Lemma 3
  • proof