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 $\left | \mathrm {Kl}(1, n)\right |$.References
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Additional Information
- Sary Drappeau
- Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France
- MR Author ID: 1021148
- Email: sary-aurelien.drappeau@univ-amu.fr
- James Maynard
- Affiliation: Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
- MR Author ID: 1007204
- Email: james.alexander.maynard@gmail.com
- 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
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3876731