Sums of three cubes, II

TL;DR

This paper improves estimates for moments of cubic Weyl sums and uses these to establish a better lower bound on the count of integers representable as sums of three cubes, advancing understanding in additive number theory.

Contribution

It introduces an enhanced iterative method for estimating moments of cubic Weyl sums, leading to improved bounds on sums of three cubes.

Findings

Enhanced estimates for moments of cubic Weyl sums for 4 ≤ s ≤ 8.

Improved lower bounds on the number of integers as sums of three cubes.

Advancement beyond classical convexity in estimating Weyl sums.

Abstract

Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\le s\le 8$, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented for the number of integers not exceeding $X$ that are represented as the sum of three cubes of natural numbers.