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Negative first moment of quadratic twists of $L$-functions

Peng Gao, Liangyi Zhao

TL;DR

The paper studies the negative first moment of quadratic twists of automorphic L-functions at $1/2+\alpha$ (with $0<\alpha\le 1/2$) under GRH and the Ramanujan-Petersson conjecture. It introduces a two-variable Dirichlet series $A(s,z;\pi)$ encoding the moment, proves its meromorphic continuation and a simple pole at $s=1$, and derives an asymptotic via Mellin inversion that features a main term $X\widehat{w}(1)P(1/2+\alpha;\pi)$ and a power-saving error. The residue $P(z;\pi)$ is given by a convergent Euler product, reflecting the arithmetic data of the automorphic representation $\pi$. The method extends the scope of negative moments for quadratic twists by employing the framework of multiple Dirichlet series, Gauss sums, and functional equations, under standard hypotheses, and provides insight into non-vanishing phenomena in this automorphic setting.

Abstract

We evaluate asymptotically the negative first moment at points larger than $1/2$ of the family of quadratic twists of automorphic $L$-functions using multiple Dirichlet series under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture.

Negative first moment of quadratic twists of $L$-functions

TL;DR

The paper studies the negative first moment of quadratic twists of automorphic L-functions at (with ) under GRH and the Ramanujan-Petersson conjecture. It introduces a two-variable Dirichlet series encoding the moment, proves its meromorphic continuation and a simple pole at , and derives an asymptotic via Mellin inversion that features a main term and a power-saving error. The residue is given by a convergent Euler product, reflecting the arithmetic data of the automorphic representation . The method extends the scope of negative moments for quadratic twists by employing the framework of multiple Dirichlet series, Gauss sums, and functional equations, under standard hypotheses, and provides insight into non-vanishing phenomena in this automorphic setting.

Abstract

We evaluate asymptotically the negative first moment at points larger than of the family of quadratic twists of automorphic -functions using multiple Dirichlet series under the generalized Riemann hypothesis and the Ramanujan-Petersson conjecture.
Paper Structure (13 sections, 7 theorems, 77 equations)

This paper contains 13 sections, 7 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadratic Gauss sums
  4. Automorphic $L$-functions
  5. Bounds on $L$-functions
  6. Functional equations for quadratic Dirichlet $L$-functions
  7. Some results on multivariable complex functions
  8. Proof of Theorem \ref{['Theorem for all characters']}
  9. First region of absolute convergence of $A(s,z;\pi)$
  10. Residue of $A(s,z;\pi)$ at $s=1$
  11. Second region of absolute convergence of $A(s,z;\pi)$
  12. Bounding $A(s,z;\pi)$ in vertical strips
  13. Completion of proof

Key Result

Theorem 1.1

Suppose that GRH and the Ramanujan-Petersson conjecture are true. Let $M \geq 1$ and let $\pi$ be a fixed self-contragredient representation of $\text{GL}_{M}$ over $\mathbb Q$ such that $\delta(\pi)=-1$. Let $w(t)$ be a non-negative Schwartz function compactly supported on $(0, \infty)$ and $\wideh where

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.7
  • Definition 2.9
  • Proposition 2.10
  • Proposition 2.11
  • Lemma 3.4
  • proof