A mean-value theorem for exponential sums. Applications to Weyl's inequality

TL;DR

This paper investigates the mean value of sixth powers of exponential sums, applying the findings to improve Weyl's inequality using ideas from Bombieri, Iwaniec, and Heath-Brown.

Contribution

It introduces a new mean-value theorem for exponential sums that enhances the understanding and application of Weyl's inequality.

Findings

Established a new mean-value estimate for exponential sums.

Applied the estimate to refine Weyl's inequality.

Demonstrated the effectiveness of Bombieri-Iwaniec and Heath-Brown techniques.

Abstract

We study the mean value of sixth power of some exponential sums following an idea due to Bombieri and Iwaniec. The result applies to Weyl's inequality, following an idea due to Heath-Brown