We prove that the existence of exceptional real zeros of Dirichlet Lfunctions would lead to cancellations in the sum p≤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 |Kl(1, n)|
