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Non-vanishing and One Level Density for Dirichlet $L$-functions Along Short Averages

Debmalya Basak

TL;DR

This paper analyzes non-vanishing and low-lying zeros of primitive Dirichlet L-functions $L(s,\chi)$ at the central point $s=\tfrac{1}{2}$ when moduli $q$ are averaged over short intervals and arithmetic progressions. The authors develop a refined mollifier approach with a two-piece mollifier that, together with exponential oscillations detecting short-range averaging, yields non-vanishing proportions exceeding $\tfrac{1}{2}-\delta$ for suitably small $\eta_1,\eta_2$ satisfying $9\eta_1+\eta_2<\tfrac{1}{16}$, and they also obtain one-level density results with extended Fourier support. The analysis hinges on precise mollified moments, harnessing deep bounds on sums of Kloosterman sums in residue classes (via Deshouille–Iwaniec and a Drappeau–Radziwiłł framework), and a dispersion approach for the density problem; a GRH assumption further pushes the non-vanishing threshold beyond $1/2$ in the short-range setting. The results extend the reach of averaged non-vanishing and density phenomena to short moduli ranges and arithmetic progressions, with explicit constants governing how far the support can be pushed and how the majorants depend on the short-range parameters. Overall, the work advances the understanding of when half-plus of central L-values are non-zero under restricted averaging and demonstrates the power of combining mollification, dispersion, and refined exponential sum bounds in this setting.

Abstract

Assuming the Generalized Riemann Hypothesis, it is known that at least half of the central values $L(\frac{1}{2},χ)$ are non-vanishing as $χ$ ranges over primitive characters modulo $q$. Unconditionally, this is known on average over both $χ$ modulo $q$ and $Q/2 \leq q \leq 2Q$. We prove that for any $δ>0$, there exist $η_1,η_2>0$ depending on $δ$ such that the non-vanishing proportion for $L(\frac{1}{2},χ)$ as $χ$ ranges modulo $q$ with $q$ varying in short intervals of size $Q^{1-η_1}$ around $Q$ and in arithmetic progressions with moduli up to $Q^{η_2}$ is larger than $\frac{1}{2}-δ$. Furthermore, by studying the one-level density of low-lying zeros of $L(s, χ)$, we show that under the Generalized Riemann Hypothesis, non-vanishing proportions exceeding $\frac{1}{2}$ can be obtained while still averaging over short ranges of $q$.

Non-vanishing and One Level Density for Dirichlet $L$-functions Along Short Averages

TL;DR

This paper analyzes non-vanishing and low-lying zeros of primitive Dirichlet L-functions at the central point when moduli are averaged over short intervals and arithmetic progressions. The authors develop a refined mollifier approach with a two-piece mollifier that, together with exponential oscillations detecting short-range averaging, yields non-vanishing proportions exceeding for suitably small satisfying , and they also obtain one-level density results with extended Fourier support. The analysis hinges on precise mollified moments, harnessing deep bounds on sums of Kloosterman sums in residue classes (via Deshouille–Iwaniec and a Drappeau–Radziwiłł framework), and a dispersion approach for the density problem; a GRH assumption further pushes the non-vanishing threshold beyond in the short-range setting. The results extend the reach of averaged non-vanishing and density phenomena to short moduli ranges and arithmetic progressions, with explicit constants governing how far the support can be pushed and how the majorants depend on the short-range parameters. Overall, the work advances the understanding of when half-plus of central L-values are non-zero under restricted averaging and demonstrates the power of combining mollification, dispersion, and refined exponential sum bounds in this setting.

Abstract

Assuming the Generalized Riemann Hypothesis, it is known that at least half of the central values are non-vanishing as ranges over primitive characters modulo . Unconditionally, this is known on average over both modulo and . We prove that for any , there exist depending on such that the non-vanishing proportion for as ranges modulo with varying in short intervals of size around and in arithmetic progressions with moduli up to is larger than . Furthermore, by studying the one-level density of low-lying zeros of , we show that under the Generalized Riemann Hypothesis, non-vanishing proportions exceeding can be obtained while still averaging over short ranges of .
Paper Structure (37 sections, 21 theorems, 269 equations)

This paper contains 37 sections, 21 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main Results
  4. Outline of the Proofs
  5. Structure of the Paper
  6. Notations
  7. Proof of Theorem \ref{['Main Theorem']}: Initial Setup
  8. A Survey of the Mollifier Method
  9. The Mollified Moments
  10. Bounds on Sums of Kloosterman Sums in Residue Classes
  11. Proof of Theorem \ref{['Main Theorem']}: Preliminary Reductions
  12. Localizing Parameters
  13. Separation of Variables
  14. Character Orthogonality and Range Reductions
  15. Reducing the range of $v$.
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\eta_1, \eta_2 \geq 0$ be fixed such that Then there exists a constant $c(\eta_1, \eta_2) > 0$ depending only on $\eta_1$ and $\eta_2$ with the following property. Let $\Psi$ be a fixed nonnegative smooth function compactly supported in $[\frac{1}{2},\frac{3}{2}]$ with $\Psi(1) > 0$. Consider $a,D \textrm{ and } Q\in \mathbb{N}$ such that $ where $\mathcal{H}_{Q^{\eta_1}}(t)$ is defined by D

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • ...and 30 more