Bounds on bilinear forms with Kloosterman sums

TL;DR

This paper presents improved bounds on bilinear forms involving Kloosterman sums, utilizing new estimates and additive combinatorics techniques, with applications to moments of L-functions and divisor function distributions.

Contribution

It introduces novel bounds on bilinear forms with Kloosterman sums and develops new methods using additive combinatorics over prime fields.

Findings

Enhanced bounds for bilinear forms with Kloosterman sums.

Application of new estimates to moments of Dirichlet L-functions.

Results on divisor function distribution in arithmetic progressions.

Abstract

We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by \'E. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, \'E. Fouvry, E. Kowalski, Ph. Michel and D. Mili\'cevi\'c (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet $L$-functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.