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Bilinear forms with Kloosterman sums and moments of twisted L-functions

Djordje Milićević, Xinhua Qin, Xiaosheng Wu

Abstract

We establish power-saving estimates for general bilinear forms with Kloosterman sums modulo arbitrary q, including when both variables are shorter than the Polya-Vinogradov range. As an application, we obtain power-saving asymptotics for the second moment of (holomorphic or Maass) modular L-functions twisted with Dirichlet characters to an arbitrary large admissible modulus q. The bounds obtained are independent of the Ramanujan-Petersson conjecture and remove all factorability conditions on q in the work of Blomer, Fouvry, Kowalski, Michel, Milicevic, and Sawin.

Bilinear forms with Kloosterman sums and moments of twisted L-functions

Abstract

We establish power-saving estimates for general bilinear forms with Kloosterman sums modulo arbitrary q, including when both variables are shorter than the Polya-Vinogradov range. As an application, we obtain power-saving asymptotics for the second moment of (holomorphic or Maass) modular L-functions twisted with Dirichlet characters to an arbitrary large admissible modulus q. The bounds obtained are independent of the Ramanujan-Petersson conjecture and remove all factorability conditions on q in the work of Blomer, Fouvry, Kowalski, Michel, Milicevic, and Sawin.

Paper Structure

This paper contains 28 sections, 21 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Bilinear forms with Kloosterman sums
  3. Moments of twisted $L$-functions
  4. Notation
  5. Sketching the treatment for bilinear forms
  6. Estimates on bilinear forms in the factorable and rough regimes
  7. Proof of Theorem \ref{['thmmain']}
  8. Sketch of the treatments for Theorems \ref{['thmBKs']} & \ref{['thmBK']}
  9. Bilinear forms for well-factorable moduli
  10. The case for high powers of $2$
  11. Generalities about the $2$-adic square root
  12. Kloosterman sum evaluation
  13. Estimation of complete sums
  14. Proof of Theorem \ref{['lemSumMK']}
  15. Bilinear forms for moduli with large prime factors
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $q$ be a positive integer, $M,N\ge1$, and let $\bm{\alpha}=(\alpha_m)$, $\bm{\beta}=(\beta_n)$ be two sequences supported respectively on $[1,M]$ and $[1,N]$. If the conditions are satisfied, then for any integer $c$ coprime with $q$, we have

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['thmmain']}
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 31 more