Visual properties of generalized Kloosterman sums

Paula Burkhardt; Alice Zhuo Yu Chan; Gabriel Currier; Stephan Ramon Garcia; Florian Luca; Hong Suh

doi:10.1016/j.jnt.2015.08.019

Visual properties of generalized Kloosterman sums

Paula Burkhardt
, Alice Zhuo Yu Chan
, Gabriel Currier
, Stephan Ramon Garcia
, Florian Luca
, Hong Suh

Mathematics

Stony Brook University

University of California at San Diego
Pomona College
University of the Witwatersrand

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

For a positive integer m and a subgroup Λ of the unit group (Z/mZ)×, the corresponding generalized Kloosterman sum is the function K(a,b,m,Λ)=∑u∈Λe(au+bu-1m) for a,b∈Z/mZ. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.

Original language	English
Pages (from-to)	237-253
Number of pages	17
Journal	Journal of Number Theory
Volume	160
DOIs	https://doi.org/10.1016/j.jnt.2015.08.019
State	Published - Mar 1 2016

Keywords

Equidistribution
Gauss sum
Hypocycloid
Kloosterman sum
Lucas number
Lucas prime
Salié sum
Supercharacter
Uniform distribution

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10.1016/j.jnt.2015.08.019

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title = "Visual properties of generalized Kloosterman sums",

abstract = "For a positive integer m and a subgroup Λ of the unit group (Z/mZ)×, the corresponding generalized Kloosterman sum is the function K(a,b,m,Λ)=∑u∈Λe(au+bu-1m) for a,b∈Z/mZ. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.",

keywords = "Equidistribution, Gauss sum, Hypocycloid, Kloosterman sum, Lucas number, Lucas prime, Sali{\'e} sum, Supercharacter, Uniform distribution",

author = "Paula Burkhardt and Chan, \{Alice Zhuo Yu\} and Gabriel Currier and Garcia, \{Stephan Ramon\} and Florian Luca and Hong Suh",

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doi = "10.1016/j.jnt.2015.08.019",

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AU - Burkhardt, Paula

AU - Chan, Alice Zhuo Yu

AU - Currier, Gabriel

AU - Garcia, Stephan Ramon

AU - Luca, Florian

AU - Suh, Hong

PY - 2016/3/1

Y1 - 2016/3/1

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AB - For a positive integer m and a subgroup Λ of the unit group (Z/mZ)×, the corresponding generalized Kloosterman sum is the function K(a,b,m,Λ)=∑u∈Λe(au+bu-1m) for a,b∈Z/mZ. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.

KW - Equidistribution

KW - Gauss sum

KW - Hypocycloid

KW - Kloosterman sum

KW - Lucas number

KW - Lucas prime

KW - Salié sum

KW - Supercharacter

KW - Uniform distribution

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DO - 10.1016/j.jnt.2015.08.019

M3 - Article

SN - 0022-314X

VL - 160

SP - 237

EP - 253

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

Visual properties of generalized Kloosterman sums

Abstract

Keywords

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