For a positive integer m and a subgroup Λ of the unit group (Z/m Z)×, the corresponding
generalized Kloosterman sum is the function K (a, b, m, Λ)=∑ u∈ Λ e (a u+ bu− 1 m) for a,
b∈ Z/m Z. 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.

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.