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Non-Vanishing of Dirichlet $L$-functions at the central point with restricted root number

Adam Earnst

Abstract

We prove asymptotics for mollified first and second moments of subfamilies of Dirichlet $L$-functions given by shrinking angular restrictions on the root number. Using these moments, we prove that for even primitive characters with prime conductor $q$, a positive proportion of the central values $L(1/2,χ)$ do not vanish as $q\to\infty$.

Non-Vanishing of Dirichlet $L$-functions at the central point with restricted root number

Abstract

We prove asymptotics for mollified first and second moments of subfamilies of Dirichlet -functions given by shrinking angular restrictions on the root number. Using these moments, we prove that for even primitive characters with prime conductor , a positive proportion of the central values do not vanish as .
Paper Structure (15 sections, 14 theorems, 151 equations)

This paper contains 15 sections, 14 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Remarks
  4. Acknowledgments
  5. Setup
  6. Weighted moments
  7. Weighted first moments
  8. Weighted second moments
  9. Bounds on for the $k=1$ case
  10. Mollified moments
  11. Mollified first moments
  12. Mollified second moment
  13. Smoothed moments
  14. Non-Vanishing
  15. Evaluation of an arithmetic sum

Key Result

Theorem 1.1

Let $\{I_q\}$ be a sequence of intervals varying over $q$ prime. Suppose that $\mu(I_q)\geq C q^{-\eta}$ for some $0\leq\eta<1/480$. Then, for $q$ sufficiently large depending on $C$, $\eta$, and $\epsilon$, where $\varphi^+(q)$ denotes the number of primitive even characters modulo $q$ and $c(\eta)$ is positive.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 2.1
  • Theorem 2.2: Approximate functional equation, IK04, Theorem 5.3
  • Remark 2.3
  • ...and 11 more