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Zeros of Dirichlet $L$-functions on the critical line

Keiju Sono

Abstract

In this paper, we estimate the proportion of zeros of Dirichlet $L$-functions on the critical line. Using Feng's mollifier and an asymptotic formula for the mean square of Dirichlet $L$-functions, we prove that averaged over primitive characters and conductors, at least 61.07 % of zeros of Dirichlet $L$-functions are on the critical line, and at least 60.44 % of zeros are simple and on the critical line. These results improve the work of Conrey, Iwaniec and Soundararajan.

Zeros of Dirichlet $L$-functions on the critical line

Abstract

In this paper, we estimate the proportion of zeros of Dirichlet -functions on the critical line. Using Feng's mollifier and an asymptotic formula for the mean square of Dirichlet -functions, we prove that averaged over primitive characters and conductors, at least 61.07 % of zeros of Dirichlet -functions are on the critical line, and at least 60.44 % of zeros are simple and on the critical line. These results improve the work of Conrey, Iwaniec and Soundararajan.

Paper Structure

This paper contains 6 sections, 20 theorems, 213 equations, 1 table.

Table of Contents

  1. Introduction
  2. Evaluating the proportion of critical zeros
  3. The adaption of Feng's mollifier
  4. Completion of proofs of theorems
  5. Appendix. The validity of (\ref{['31']}) with (\ref{['32.5']}) in a larger domain
  6. Acknowledgements

Key Result

Theorem 1.1

For $(\log Q)^{2} \leq T \leq (\log Q)^{A}$, we have and where $A >2$ is an arbitrary fixed constant and $Q$ is sufficiently large in terms of $A$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2: Conrey, Iwaniec, Soundararajan CIS3
  • Theorem 1.3
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2: F, (3.27)
  • Lemma 3.3
  • proof
  • Lemma 3.4: L, Lemma 3.9
  • ...and 17 more