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Non-vanishing of Dirichlet $L$-functions with smooth conductors

Sun-Kai Leung

TL;DR

The paper addresses the central-value non-vanishing problem for primitive Dirichlet L-functions with smooth conductors, proving that for large square-free moduli $q$ that are $q^{\eta}$-smooth, at least $35.9\%$ of characters satisfy $L\left(\tfrac{1}{2},\chi\right)\neq 0$. The authors employ an unbalanced two-piece mollifier and analyze mollified first and second moments via Cauchy–Schwarz, Poisson summation on smooth moduli, and a $q$-van der Corput AB-process within the trace-function framework, achieving an explicit arithmetic-exponent-pair $(\kappa,\lambda,\nu)=(\tfrac{52}{243},\tfrac{50}{81},\tfrac{202}{243})$. This yields the main asymptotics for the mollified second moment and, consequently, a lower bound on the non-vanishing proportion. The work also discusses limitations, noting that unconditional barriers near $40\%$ remain, with conditional improvements possible under arithmetic-exponent-pair hypotheses and potential extensions with more mollifier pieces.

Abstract

Given a large, square-free, smooth conductor, we establish the non-vanishing of the central values for at least $35.9\%$ of the primitive Dirichlet $L$-functions.

Non-vanishing of Dirichlet $L$-functions with smooth conductors

TL;DR

The paper addresses the central-value non-vanishing problem for primitive Dirichlet L-functions with smooth conductors, proving that for large square-free moduli that are -smooth, at least of characters satisfy . The authors employ an unbalanced two-piece mollifier and analyze mollified first and second moments via Cauchy–Schwarz, Poisson summation on smooth moduli, and a -van der Corput AB-process within the trace-function framework, achieving an explicit arithmetic-exponent-pair . This yields the main asymptotics for the mollified second moment and, consequently, a lower bound on the non-vanishing proportion. The work also discusses limitations, noting that unconditional barriers near remain, with conditional improvements possible under arithmetic-exponent-pair hypotheses and potential extensions with more mollifier pieces.

Abstract

Given a large, square-free, smooth conductor, we establish the non-vanishing of the central values for at least of the primitive Dirichlet -functions.
Paper Structure (4 sections, 6 theorems, 59 equations)

This paper contains 4 sections, 6 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Unbalanced two-piece mollifier
  3. Mollified moments
  4. Limitation

Key Result

Theorem 1.1

There exist positive real numbers $Q, \eta$ such that for any square-free integer $q>Q$ which is $q^{\eta}$-smooth, i.e., without prime factors exceeding $q^{\eta}$, we have $L(\frac{1}{2},\chi) \neq 0$ for at least $35.9\%$ of the primitive characters modulo $q.$

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 2.1: Mollified first moment
  • Definition 2.1: Arithmetic exponent pair MR4355471
  • Proposition 2.2: Mollified second moment
  • proof : Proof of Theorem \ref{['thm:main']} assuming Propositions \ref{['prop:1st']} and \ref{['prop:2nd']}
  • proof : Proof of Proposition \ref{['prop:1st']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 4 more