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Twisted first moment of quadratic and quadratic twist $L$-functions

Peng Gao, Liangyi Zhao

Abstract

We evaluate the twisted first moment of central values of the product of a quadratic Dirichlet $L$-function and a quadratic twist of a modular $L$-function.

Twisted first moment of quadratic and quadratic twist $L$-functions

Abstract

We evaluate the twisted first moment of central values of the product of a quadratic Dirichlet -function and a quadratic twist of a modular -function.
Paper Structure (14 sections, 10 theorems, 120 equations)

This paper contains 14 sections, 10 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sums over primes
  4. Modular $L$-functions
  5. The approximate functional equation
  6. Gauss sums and Poisson summation
  7. Upper bounds for moments of quadratic twists of $L$-functions
  8. Analytic behaviors of some Dirichlet Series
  9. Proof of Theorem \ref{['theo:1stmoment']}
  10. A first decomposition
  11. Estimating $S_2(l)$
  12. Estimating $S_1(l)$, the main term
  13. Estimating $S_1(l)$, the $k \neq 0$ terms
  14. Completion of the proof

Key Result

Theorem 1.1

With the notation as above and $\kappa \equiv 0 \pmod 4$, let $\Phi:\mathbb R^{+} \rightarrow \mathbb R$ be a smooth function of compact support. For any positive odd integer $l$ such that $l=l_1l^2_2$ with $l_1$ square-free, suppose that for any $p|l_1$, Then we have for any $\varepsilon>0$, where the functions $\mathcal{H}(s)$ and $Q_p(s;l)$ are given in Lemma lemma:DS4P below. Moreover, $C$ i

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.2
  • proof
  • Lemma 2.5: Approximate functional equation
  • proof
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.10
  • proof
  • ...and 6 more