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arXiv · 1802.10278

Sign changes of Kloosterman sums and exceptional characters

Sary Drappeau·James Maynard
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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 $\left| \Kl(1, n)\right|$.

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Sary Drappeau, James Maynard. 2018-02-28. Sign changes of Kloosterman sums and exceptional characters. https://doi.org/10.1090/proc%2F14239

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