Lower bounds on odd order character sums

TL;DR

This paper establishes optimal lower bounds on character sums for odd order characters, extending classical results and matching recent upper bounds, thus deepening understanding of character sum behavior.

Contribution

It provides the first known lower bounds for odd order character sums that align with recent upper bounds, generalizing classical quadratic character results.

Findings

Lower bounds on odd order character sums are established.

Results match recent conditional upper bounds, confirming their optimality.

Extends classical quadratic character sum results to odd order characters.

Abstract

A classical result of Paley shows that there are infinitely many quadratic characters $\chi\mod{q}$ whose character sums get as large as $\sqrt{q}\log \log q$; this implies that a conditional upper bound of Montgomery and Vaughan cannot be improved. In this paper, we derive analogous lower bounds on character sums for characters of odd order, which are best possible in view of the corresponding conditional upper bounds recently obtained by the first author.