Large even order character sums

TL;DR

This paper extends Paley's classical theorem to characters of any fixed even order, demonstrating the existence of large character sums without relying on the Generalized Riemann Hypothesis.

Contribution

It provides an unconditional proof of large character sums for even order characters, generalizing previous results limited to quadratic characters.

Findings

Existence of large character sums for even order characters

Unconditional results without GRH assumption

Extension of classical theorems to broader character classes

Abstract

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order. Previously our bounds were only known under the assumption of the Generalized Riemann Hypothesis.