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Kloosterman paths and the shape of exponential sums

Part of: Exponential sums and character sums Stochastic processes (Co)homology theory Limit theorems Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  15 April 2016

Emmanuel Kowalski  and
William F. Sawin
Emmanuel Kowalski
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, CH-8092 Zürich, Switzerland email kowalski@math.ethz.ch
William F. Sawin
Affiliation:
Princeton University, Fine Hall, Washington Road, NJ, USA email wsawin@math.princeton.edu
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Abstract

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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.

Keywords

Kloosterman sumsKloosterman sheavesRiemann Hypothesis over finite fieldsrandom Fourier seriesshort exponential sumsprobability in Banach spaces

MSC classification

Primary: 11T23: Exponential sums 11L05: Gauss and Kloosterman sums; generalizations 14F20: Étale and other Grothendieck topologies and (co)homologies 60F17: Functional limit theorems; invariance principles 60G17: Sample path properties
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 7 , July 2016 , pp. 1489 - 1516
Copyright
© The Authors 2016 

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