Graphical cyclic supercharacters for composite moduli

TL;DR

This paper explores the graphical properties of cyclic supercharacters for composite moduli, extending previous work focused on prime powers to more general composite numbers, revealing new visual patterns and properties.

Contribution

It introduces the study of graphical images of cyclic supercharacters for composite moduli, expanding understanding beyond prime power cases.

Findings

Revealed new graphical patterns for composite moduli

Extended prior prime power results to general composite numbers

Connected supercharacter plots to exponential sums and Gaussian periods

Abstract

Recent work has introduced the study of graphical properties of cyclic supercharacters, functions $\mathbb{Z}/n\mathbb{Z}\to \mathbb{C}$ whose values are exponential sums with close connections to Gauss sums and Gaussian periods. Plots of these functions exhibit striking features, some of which have been previously explained when the modulus $n$ is a power of an odd prime. After reviewing this material, we initiate the graphical study of images of cyclic supercharacters in the case of composite $n$.