A note on exponential-M\"{o}bius sums over $\mathbb{F}_q[t]$

TL;DR

This paper extends a classical exponential sum bound from integers to polynomials over finite fields by leveraging Weil's Riemann Hypothesis, providing a new perspective on exponential sums in algebraic function fields.

Contribution

It derives an analogous exponential sum bound over finite fields, adapting methods from Hayes and Weil's Riemann Hypothesis for curves.

Findings

Established a bound on exponential sums over $F_q[t]$

Connected classical integer results to finite field analogs

Utilized Weil's Riemann Hypothesis for curves

Abstract

In 1991, Baker and Harman proved, under the assumption of the generalized Riemann hypothesis, that $\max_{ \theta \in [0,1) }\left|\sum_{ n \leq x } \mu(n) e(n \theta) \right| \ll_\epsilon x^{3/4 + \epsilon}$. The purpose of this note is to deduce an analogous bound in the context of polynomials over a finite field using Weil's Riemann Hypothesis for curves over a finite field. Our approach is based on the work of Hayes who studied exponential sums over irreducible polynomials.