Visual properties of generalized Kloosterman sums

Authors

Paula Burkhardt '16, Pomona College
Alice Zhuo-Yu Chan '14, Pomona College
Gabriel Currier '16, Pomona College
Stephan Ramon Garcia, Pomona CollegeFollow
Florian Luca, University of Witwatersrand
Hong Suh '16, Pomona College

Student Co-author

Pomona Undergraduate

Document Type

Article - preprint

Department

Mathematics (Pomona)

Publication Date

2016

Keywords

Kloosterman sum, Gauss sum, Salie sum, supercharacter, hypocycloid, uniform distribution, equidistribution, Lucas number, Lucas prime

Abstract

For a positive integer m and a subgroup A of the unit group (Z/mZ)^x, the corresponding generalized Kloosterman sum is the function K(a, b, m, A) = Σ_uEA e(au+bu^-1/m). 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.

Comments

Final published version can be found at: Burkhardt, P., Chan, A. Z.-Y., Currier, G., Garcia, S.R., Luca, F., Suh, H., Visual properties of generalized Kloosterman sums, Journal of Number Theory 160 (2016), 237-253.

Rights Information

© 2015 Elsevier Inc.

DOI

10.1016/j.jnt.2015.08.019

Recommended Citation

Burkhardt, P., Chan, A. Z.-Y., Currier, G., Garcia, S.R., Luca, F., Suh, H., Visual properties of generalized Kloosterman sums, Journal of Number Theory 160 (2016), 237-253.