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Sign changes of Kloosterman sums and exceptional characters


Authors: Sary Drappeau and James Maynard
Journal: Proc. Amer. Math. Soc. 147 (2019), 61-75
MSC (2010): Primary 11L05, 11N36; Secondary 11N75, 11L20, 11M20
DOI: https://doi.org/10.1090/proc/14239
Published electronically: October 3, 2018
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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} \textup {Kl}(1, p)$ of Kloosterman sums over primes, and also to sign changes of  $ \textup {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 \vert \textup {Kl}(1, n)\right \vert$.


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Additional Information

Sary Drappeau
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France
Email: sary-aurelien.drappeau@univ-amu.fr

James Maynard
Affiliation: Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
Email: james.alexander.maynard@gmail.com

DOI: https://doi.org/10.1090/proc/14239
Received by editor(s): March 2, 2018
Received by editor(s) in revised form: March 16, 2018
Published electronically: October 3, 2018
Additional Notes: Part of this work was done during a visit of the second author to Aix-Marseille University, supported by the French-Austrian joint project MuDeRa (FWF I-1751-N26, ANR-14-CE34-0009).
Communicated by: Amanda Folsom
Article copyright: © Copyright 2018 American Mathematical Society

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