Sign changes of Kloosterman sums and exceptional characters

Drappeau, Sary; Maynard, James

doi:10.1090/proc/14239

Sign changes of Kloosterman sums and exceptional characters
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by Sary Drappeau and James Maynard PDF

Proc. Amer. Math. Soc. 147 (2019), 61-75 Request permission

Abstract:

We prove that the existence of exceptional real zeros of Dirichlet $L$-functions would lead to cancellations in the sum $\sum _{p\leq x} \mathrm {Kl}(1, p)$ of Kloosterman sums over primes, and also to sign changes of $\mathrm {Kl}(1, n)$, 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 $\left | \mathrm {Kl}(1, n)\right |$.

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