Mock characters and the Kronecker symbol

TL;DR

This paper introduces 'mock characters', a new family of multiplicative functions closely resembling Dirichlet characters, and explores their properties alongside new results on the Kronecker symbol.

Contribution

It defines and analyzes mock characters, revealing their properties and their near equivalence to Dirichlet characters, along with new insights into the Kronecker symbol's behavior.

Findings

Mock characters closely resemble Dirichlet characters.

New properties of the Kronecker symbol are established.

Mock characters exhibit unique multiplicative properties.

Abstract

We introduce and study a family of functions we call the "mock characters". These functions satisfy a number of interesting properties, and of all completely multiplicative arithmetic functions seem to come as close as possible to being Dirichlet characters. Along the way we prove a few new results concerning the behavior of the Kronecker symbol.