Visual properties of generalized Kloosterman sums

Abstract

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

14 pages

References in corpus (1)

• Mean eigenvalues for simple, simply connected, compact Lie groups

Cited by in corpus (1)

• Graphical cyclic supercharacters for composite moduli