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Kloosterman paths and the shape of exponential sums

Published online by Cambridge University Press:  15 April 2016

Emmanuel Kowalski
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, CH-8092 Zürich, Switzerland email [email protected]
William F. Sawin
Affiliation:
Princeton University, Fine Hall, Washington Road, NJ, USA email [email protected]

Abstract

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums $\text{Kl}_{p}(a)$, as $a$ varies over $\mathbf{F}_{p}^{\times }$ and as $p$ tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

Type
Research Article
Copyright
© The Authors 2016 

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