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Conditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros

Tianyu Zhao

Abstract

Under the generalized Riemann hypothesis, we use Beurling-Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet $L$-functions to a large prime modulus $q$. As applications, we give alternative proofs of several results on low-lying zeros of $L(s,χ)$ and obtain a new lower bound on the proportion of $L(s,χ)$ modulo $q$ with zeros close to the central point $s=1/2$. In particular, we show conditionally that for any $β>1/4$, there exist a positive proportion of Dirichlet $L$-functions whose first zero has height less than $β$ times the average spacing between consecutive zeros.

Conditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros

Abstract

Under the generalized Riemann hypothesis, we use Beurling-Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet -functions to a large prime modulus . As applications, we give alternative proofs of several results on low-lying zeros of and obtain a new lower bound on the proportion of modulo with zeros close to the central point . In particular, we show conditionally that for any , there exist a positive proportion of Dirichlet -functions whose first zero has height less than times the average spacing between consecutive zeros.

Paper Structure

This paper contains 7 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preparations and lemmas
  3. Proofs of Theorems \ref{['theorem 1']} and \ref{['theorem 2']}
  4. Proof of Theorem \ref{['theorem 1']}
  5. Proof of Theorem \ref{['theorem 2']}
  6. Further remarks
  7. Acknowledgment

Key Result

Theorem 1

Assuming GRH, where with the implied constant being uniform in $q$ and $T>0$. In particular, as $q\to \infty$, $|\mathbb{E}[\widetilde{S}(T,\chi)]|\leq 1/2+o(1)$ if $T\ll q^{o(1)}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (14)

  • Theorem 1
  • Remark 1
  • Corollary 1
  • proof
  • Theorem 2
  • Corollary 2
  • proof : Proof of Corollary \ref{['corollary: proportion']}
  • Lemma 3
  • Lemma 4
  • proof
  • ...and 4 more