Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Twisted second moment of modular $L$-functions to a fixed modulus

Peng Gao, Liangyi Zhao

TL;DR

The paper addresses the problem of evaluating the twisted second moment for the family of modular $L$-functions $L(s,f\otimes\chi)$ with a fixed modulus $q$, and uses this to derive sharp bounds for the $2k$-th moments on the critical line. The authors derive an explicit asymptotic formula for the twisted second moment via a diagonal/off-diagonal analysis, employing an approximate functional equation, Voronoi summation, and Kloosterman-sum bounds, leading to a main term involving $\zeta(s_1+s_2)L(s_1+s_2,\operatorname{sym}^2 f)$ and a local factor $H(s_1+s_2; q,a,b)$. They then apply lower- and upper-bound principles to obtain sharp lower bounds for all $k\ge 0$ and sharp upper bounds for $0\le k\le 1$ for the $2k$-th moment, yielding asymptotics matching up to constants $\varphi^*(q)(\log q)^{k^2}$. This work advances understanding of level-aspect moments of modular $L$-functions and provides tools for precise moment control on the critical line. The results have potential implications for related moment problems and character-sum estimates in the fixed-modulus setting.

Abstract

We study asymptotically the twisted second moment of the family of modular $L$-functions to a fixed modulus. As an application, we establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$ for the $2k$-th moment of these $L$-functions on the critical line.

Twisted second moment of modular $L$-functions to a fixed modulus

TL;DR

The paper addresses the problem of evaluating the twisted second moment for the family of modular -functions with a fixed modulus , and uses this to derive sharp bounds for the -th moments on the critical line. The authors derive an explicit asymptotic formula for the twisted second moment via a diagonal/off-diagonal analysis, employing an approximate functional equation, Voronoi summation, and Kloosterman-sum bounds, leading to a main term involving and a local factor . They then apply lower- and upper-bound principles to obtain sharp lower bounds for all and sharp upper bounds for for the -th moment, yielding asymptotics matching up to constants . This work advances understanding of level-aspect moments of modular -functions and provides tools for precise moment control on the critical line. The results have potential implications for related moment problems and character-sum estimates in the fixed-modulus setting.

Abstract

We study asymptotically the twisted second moment of the family of modular -functions to a fixed modulus. As an application, we establish sharp lower bounds for all real and sharp upper bounds for in the range for the -th moment of these -functions on the critical line.

Paper Structure

This paper contains 12 sections, 12 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cusp form $L$-functions
  4. The approximate functional equation
  5. Proofs of Theorem \ref{['thmtwistedsecondmoment']}
  6. Diagonal terms
  7. Off-diagonal terms
  8. Conclusion
  9. Proofs of Theorem \ref{['thmlowerbound']}--\ref{['thmupperbound']}
  10. Initial Treatments
  11. Proof of Proposition \ref{['Prop4']}
  12. Proof of Proposition \ref{['Prop5']}

Key Result

Theorem 1.1

With the notation as above, let $q \not \equiv 2 \pmod 4$ be a positive integer, $a, b$ be positive integers such that $(a,b)=(ab, q)=1$ and $s_1=\sigma_1+it_1$ and $s_2=\sigma_2+it_2$ with $0<\sigma_1$, $\sigma_2<1$, $t_1, t_2\in \mathbb R$, $s_1+s_2 \neq 1$. Suppose that one of the two conditions Suppose moreover corresponding to the above two conditions, Then we have, where $H(s; q, a, b)= \

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.3
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • ...and 3 more