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Average non-vanishing of Dirichlet $L$-functions at the central point

Kyle Pratt

Abstract

The Generalized Riemann Hypothesis implies that at least 50% of the central values $L \left( \frac{1}{2},χ\right)$ are non-vanishing as $χ$ ranges over primitive characters modulo $q$. We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo $q$ and averages $q$ over an interval, then at least 50.073% of the central values are non-vanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec. Note: The author has been made aware of an error in this work. It seems the error can be fixed, by using a different argument, and the author will present a correction in due course.

Average non-vanishing of Dirichlet $L$-functions at the central point

Abstract

The Generalized Riemann Hypothesis implies that at least 50% of the central values are non-vanishing as ranges over primitive characters modulo . We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo and averages over an interval, then at least 50.073% of the central values are non-vanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec. Note: The author has been made aware of an error in this work. It seems the error can be fixed, by using a different argument, and the author will present a correction in due course.

Paper Structure

This paper contains 9 sections, 11 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Mollification, and a sketch for Theorem \ref{['thm: main theorem']}
  3. Proof of Theorem \ref{['thm: main theorem']}: first steps
  4. Lemma \ref{['lem: main term IS MV']} main term
  5. Lemma \ref{['lem: main term IS MV']}: error term
  6. Lemma \ref{['lem: main term B MV']}: main term
  7. First case: $c \ell m = 1$
  8. Lemma \ref{['lem: main term B MV']}: error term
  9. Acknowledgments

Key Result

Theorem 1.1

Let $\Psi$ be a fixed, nonnegative smooth function, compactly supported in $[\frac{1}{2},2]$, which satisfies Then for $Q$ sufficiently large we have

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['thm: main theorem']}
  • proof : Proof of Lemma \ref{['lem: eval of S1']}
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 9 more