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The fourth moment of quadratic Dirichlet $L$-Functions II

Quanli Shen, Joshua Stucky

Abstract

We prove an asymptotic formula with four main terms for the fourth moment of quadratic Dirichlet $L$-functions unconditionally. Our proof is based on the work of Li , Soundararajan, and Soundararajan-Young. Our proof requires several new ingredients. These include a modified large sieve estimate for quadratic characters where we consider a fourth moment, rather than a second, as well as observing cross cancellations between diagonal and off-diagonal terms, which involves somewhat delicate combinatorial arguments.

The fourth moment of quadratic Dirichlet $L$-Functions II

Abstract

We prove an asymptotic formula with four main terms for the fourth moment of quadratic Dirichlet -functions unconditionally. Our proof is based on the work of Li , Soundararajan, and Soundararajan-Young. Our proof requires several new ingredients. These include a modified large sieve estimate for quadratic characters where we consider a fourth moment, rather than a second, as well as observing cross cancellations between diagonal and off-diagonal terms, which involves somewhat delicate combinatorial arguments.
Paper Structure (38 sections, 33 theorems, 486 equations)

This paper contains 38 sections, 33 theorems, 486 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Background
  3. Outline of the Paper
  4. Notation
  5. Preliminaries
  6. Approximate Functional Equation
  7. Poisson Summation Formula for Quadratic Characters
  8. Dirichlet Series
  9. Functional Equations for Dirichlet Polynomials
  10. Smooth Functions
  11. Proof of Proposition \ref{['prop:LargeSieve']}: Initial Steps
  12. Inflation and Initial Decomposition
  13. Diagonal Contribution $\mathcal{M}^{0}$
  14. Calculation of $\mathcal{I}_1$
  15. Calculation of $\mathcal{I}_2$
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $\Phi: (0,\infty) \rightarrow \mathbb{R}$ be a smooth, compactly supported function. Let $\alpha_i\in \mathbb{C}$ and $\frac{1}{2\log X} \leq |\alpha_i| \leq \frac{1}{\log X}$ for $i=1,2,3,4$. Assume for $\eta_j=\pm 1, 0$ and $\eta_j$ not all zero. Then we have where $\lambda_i$ and $\Phi_{\{\epsilon_1,\epsilon_2,\epsilon_3,\epsilon_4\}}$ are defined in def-lambda and tranf-phi, and $\mathca

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 1.3
  • proof : Proof of Corollary \ref{['thm:firstmain']} by Theorem \ref{['thm:Main']}
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 37 more