Visual properties of generalized Kloosterman sums

JOURNAL OF NUMBER THEORY

卷:160

Article

Burkhardt, Paula²  Chan, Alice Zhuo-Yu¹  Currier, Gabriel²  Garcia, Stephan Ramon²  Luca, Florian³  Suh, Hong² 

[1] Univ Calif San Diego, Dept Math, La Jolla, CA 92093 USA

[2] Pomona Coll, Dept Math, Claremont, CA 91711 USA

[3] Univ Witwatersrand, Sch Math, ZA-2050 Johannesburg, South Africa

关键词: Kloosterman sum;    Gauss sum;    Salie sum;    Supercharacter;    Hypocycloid;    Uniform distribution;    Equidistribution;    Lucas number;    Lucas prime;   

DOI  :  10.1016/j.jnt.2015.08.019

来源: Elsevier

【 摘 要 】

For a positive integer m and a subgroup Lambda of the unit group (Z/mZ)(x), the corresponding generalized Kloosterrnan sum is the function K(a, b, m, Lambda) = Sigma(u is an element of Lambda) e(au+bu(-1)/m) for a, b is an element of 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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free