A note on Hal\'{a}sz's Theorem in $\mathbb{F}_q[t]$

TL;DR

This paper extends an improved version of Halász's Theorem, originally for integers, to the function field setting of _q[t], and investigates conditions for when the bounds are tight.

Contribution

It derives an analogous improved upper bound for Halász's Theorem in _q[t] and explores the conditions under which this bound is attained.

Findings

Derived the improved upper bound for Hale1sz's Theorem in _q[t]

Identified conditions for the bound to be tight in _q[t] setting

Extended integer results to function fields

Abstract

In the setting of the integers, Granville, Harper and Soundararajan showed that the upper bound in Hal\'{a}sz's Theorem can be improved for smoothly supported functions. We derive the analogous result for Hal\'{a}sz's Theorem in $\mathbb{F}_q[t]$, and then consider the converse question of when the general upper bound in this version of Hal\'{a}sz's Theorem is actually attained.