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

TL;DR

This paper develops new power-saving estimates for bilinear forms involving Kloosterman sums and applies these results to derive asymptotic formulas for the second moments of twisted modular L-functions, independent of the Ramanujan-Petersson conjecture.

Contribution

It introduces novel bounds for bilinear forms with Kloosterman sums that work for arbitrary moduli and short variables, enabling new results on moments of twisted L-functions.

Findings

Power-saving estimates for bilinear forms with Kloosterman sums.

Asymptotic formulas for second moments of twisted modular L-functions.

Bounds are independent of the Ramanujan-Petersson conjecture and remove factorability conditions.

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.