A note on exponential-Möbius sums over ${\mathbb{F}}_q[t]$

Abstract

In 1991, Baker and Harman proved, under the assumption of the generalized Riemann hypothesis, that ${\max }_{\theta \in [0,1)}\left|{\sum }_{n\le x}\mu (n)e(n\theta )\right|{\ll }_{\epsilon }{x}^{3/4+\epsilon }$. 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.

5 pages + references. Similar result appears in arXiv:1711.05358. Second version has stronger result due to more appropriate divisor sum bound