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The second moment of derivatives of quadratic twists of modular $L$-functions

Yujiao Jiang, Quanli Shen, Ziyang Tang

Abstract

We prove an asymptotic formula for the second moment of the first derivative of quadratic twists of modular $L$-functions with three leading order main terms. It improves the previous result of Kumar et al. with the first main term. The proof is based on the large sieve type inequality established by Li, with a key input that we convert the problem into computing an asymptotic formula for the completed twisted modular $L$-functions with large shifts.

The second moment of derivatives of quadratic twists of modular $L$-functions

Abstract

We prove an asymptotic formula for the second moment of the first derivative of quadratic twists of modular -functions with three leading order main terms. It improves the previous result of Kumar et al. with the first main term. The proof is based on the large sieve type inequality established by Li, with a key input that we convert the problem into computing an asymptotic formula for the completed twisted modular -functions with large shifts.
Paper Structure (6 sections, 10 theorems, 82 equations)

This paper contains 6 sections, 10 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Lemmas
  3. Reduce the length of Dirichlet polynomials
  4. Evaluate diagonal terms
  5. Evaluate off-diagonal terms
  6. Complete the proof

Key Result

Theorem 1.1

Assume $\kappa \equiv 2 \, (\operatorname{mod} 4)$. Let $\Phi:(0,\infty) \rightarrow \mathbb{R}$ be a smooth, compactly supported function. Then where $\sum^{*}$ denotes the sum over square-free integers. Here The factor $Z_1(0,0)$ is defined in def-Z-1, and $\widetilde{\Phi}$ is the Mellin transform of $\Phi$ defined in equ:mellin. The coefficients $c_i$, $i=1,2$ can also be calculated precisel

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • ...and 2 more