Sign changes of Kloosterman sums and exceptional characters

TL;DR

This paper links the existence of exceptional zeros of Dirichlet L-functions to sign changes and cancellations in sums of Kloosterman sums over primes and semi-primes, using advanced sieve and sum bounds.

Contribution

It establishes a connection between exceptional zeros and sign variations in Kloosterman sums, introducing new bounds and methods involving sieve techniques and prior sum estimates.

Findings

Sign changes in Kloosterman sums over integers with two prime factors.

Cancellations in sums of Kloosterman sums over primes if exceptional zeros exist.

Bounds for twisted sums of Kloosterman sums.

Abstract

We prove that the existence of exceptional real zeroes of Dirichlet $L$-functions would lead to cancellations in the sum $\sum_{p\leq x} \Kl(1, p)$ of Kloosterman sums over primes, and also to sign changes of $\Kl(1, n)$, where $n$ runs over integers with exactly two prime factors. Our arguments involve a variant of Bombieri's sieve, bounds for twisted sums of Kloosterman sums, and work of Fouvry and Michel on sums of $\left| \Kl(1, n)\right|$.