Sign changes of Kloosterman sums and exceptional characters

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 $\left| \Kl(1, n)\right|$.

11 pages

References in corpus (1)

• On a certain additive divisor problem

Cited by in corpus (3)

• Chowla and Sarnak Conjectures for Kloosterman Sums
• Exceptional characters and nonvanishing of Dirichlet $L$-functions
• Exceptional characters and prime numbers in sparse sets