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 $\left \Kl(1, n)\right$.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.10278
 Bibcode:
 2018arXiv180210278D
 Keywords:

 Mathematics  Number Theory;
 11L05;
 11N36 (Primary);
 11N75;
 11L20;
 11M20 (Secondary)
 EPrint:
 11 pages