Some applications of smooth bilinear forms with Kloosterman sums

TL;DR

This paper extends bounds on bilinear forms with Kloosterman sums, applying them to improve results in prime sums and moments of Dirichlet L-functions, advancing understanding in analytic number theory.

Contribution

It introduces an extended bound for correlation sums of Kloosterman sums with modular form coefficients, enhancing previous estimates.

Findings

Improved bounds on sums of Kloosterman sums along primes

Refined error estimates for the fourth moment of Dirichlet L-functions

Enhanced understanding of bilinear forms with Kloosterman sums

Abstract

We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet $L$-functions.