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arXiv · 1505.00018

Visual properties of generalized Kloosterman sums

Paula Burkhardt·Alice Zhuo-Yu Chan·Gabriel Currier·Stephan Ramon Garcia·Florian Luca·Hong Suh
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Abstract

For a positive integer $m$ and a subgroup $Λ$ of the unit group $(\mathbb{Z}/m\mathbb{Z})^\times$, the corresponding generalized Kloosterman sum is the function $K(a,b,m,Λ) = \sum_{u \in Λ}e(\frac{au + bu^{-1}}{m})$. 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.

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Paula Burkhardt, Alice Zhuo-Yu Chan, Gabriel Currier, Stephan Ramon Garcia, Florian Luca, Hong Suh. 2015-04-30. Visual properties of generalized Kloosterman sums. https://arxiv.org/abs/1505.00018

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