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On the second integral moment of $L$-functions

Liangxun Li

Abstract

Assume that the generalized Ramanujan conjecture holds on the automorphic $L$-function $L(s, π)$ on $GL_d$ over $\mathbb{Q}$ with $d\geq 3$, we can obtain a small log-saving non-trivial bound on the second integral moment of $L(1/2+it, π)$. Specifically the bound \[ \int_{T}^{2T}\Big|L(\frac{1}{2}+it, π)\Big|^2 d t\ll_π \frac{T^{\frac{d}{2}}}{\log^{η_d}T} \] holds for a small constant $η_d>0$. As an application, we give a new asymptotic formula for the average of the coefficient $λ_{1\boxplus π}(n)$.

On the second integral moment of $L$-functions

Abstract

Assume that the generalized Ramanujan conjecture holds on the automorphic -function on over with , we can obtain a small log-saving non-trivial bound on the second integral moment of . Specifically the bound holds for a small constant . As an application, we give a new asymptotic formula for the average of the coefficient .

Paper Structure

This paper contains 13 sections, 20 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Moments of $L$-functions on $t$-aspect
  3. The average of Dirichlet coefficients of degree $d+1$ $L$-functions
  4. Background of $L$-functions
  5. Second Moment of Dirichlet polynomial
  6. Mean value theorems
  7. Decomposition of the Dirichlet polynomial
  8. Halász type estimate
  9. Proof of Proposition \ref{['prop: 2M_estimate']}
  10. Proof of Proposition \ref{['prop: 2M_estimate2']}
  11. Proof of Theorems
  12. Proof of Theorem \ref{['thm: 2M_L']} and Theorem \ref{['thm: 2M_L2']}
  13. Proof of Theorem \ref{['thm: coef_mean']} and Theorem \ref{['thm: coef_mean2']}

Key Result

Theorem 1.1

Let $\mathbb{A}$ be the ring of adelés of $\mathbb{Q}$ and $\pi$ be an irreducible cuspidal automorphic representation of $\operatorname{GL}_d(\mathbb{A})$ with its degree $d\geq 3$. Assume that the generalized Ramanujan conjecture (GRC) holds on $L(s, \pi)$, then for $T\geq 2$ large, we have for any $0<\eta_d\leq \frac{1}{400d^4}$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 25 more