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Non-vanishing of Dirichlet $L$-functions at the Central Point

Xinhua Qin, Xiaosheng Wu

TL;DR

The paper proves that for sufficiently large moduli $q$, a positive proportion $\frac{7}{19}-\varepsilon$ of primitive Dirichlet characters $\chi$ satisfy $L\left(\tfrac12,\chi\right)\neq 0$. It advances the mollifier program by integrating a twisted mollifier, extending its effective length through an averaged analysis of a quartic Kloosterman-sum analogue $G_q$ for general moduli. A key technical contribution is Theorem le:1, which gives a robust average bound for $G_q$ over quadruples $(b_1,b_2,b_3,b_4)$, enabling longer mollifiers even when $q$ is not prime. The approach combines a refined treatment of mollified second moments with averaged exponential-sum techniques and trace-function ideas, improving prior general-modulus non-vanishing results and advancing the understanding of mollifier methods in this broader setting.

Abstract

We prove that for at least $\frac{7}{19}$ of the primitive Dirichlet characters $χ$ with large general modulus, the central value $L(\frac12,χ)$ is non-vanishing.

Non-vanishing of Dirichlet $L$-functions at the Central Point

TL;DR

The paper proves that for sufficiently large moduli , a positive proportion of primitive Dirichlet characters satisfy . It advances the mollifier program by integrating a twisted mollifier, extending its effective length through an averaged analysis of a quartic Kloosterman-sum analogue for general moduli. A key technical contribution is Theorem le:1, which gives a robust average bound for over quadruples , enabling longer mollifiers even when is not prime. The approach combines a refined treatment of mollified second moments with averaged exponential-sum techniques and trace-function ideas, improving prior general-modulus non-vanishing results and advancing the understanding of mollifier methods in this broader setting.

Abstract

We prove that for at least of the primitive Dirichlet characters with large general modulus, the central value is non-vanishing.

Paper Structure

This paper contains 15 sections, 13 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. The mollifier method of Michel and VanderKam
  3. New inputs by Khan, Milićević, and Ngo
  4. Our main results
  5. Notation
  6. The proof of Theorem $1$
  7. The mollified second moment
  8. Initial treatment of $G_{q}$
  9. Some analogues of Kloosterman sums
  10. Proof of Theorem $\ref{['le:1']}$
  11. Initial treatment of $A(q)$
  12. The bound of $A_1(q)$
  13. An upper bound for ${\mathcal{A}}_3$
  14. Estimating $A_1(q)$
  15. The bound of $A_2(q)$

Key Result

Theorem 1.1

For sufficiently large integers $q$, there are at least $\frac{7}{19}-\varepsilon$ of primitive Dirichlet characters $\chi$ (mod q) for which $L(s,\chi)\neq0$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof : Proof of Theorem $\ref{['1.1']}$
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 15 more