On the Waring--Goldbach problem for eighth and higher powers

TL;DR

This paper leverages recent advances in Vinogradov's mean value theorem to improve bounds on the Waring--Goldbach problem for powers $k \\ge 8$, notably surpassing classical results from the 1940s.

Contribution

It introduces sharper bounds for the function $H(k)$ in the Waring--Goldbach problem for all $k \\ge 8$, using new exponential sum estimates.

Findings

Established that $H(k) \\le (4k-2)\\log k + k - 7$ for large $k$

First improvement on Hua's classical bound from the 1940s

Applied recent exponential sum estimates to Waring--Goldbach problem

Abstract

Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the function $H(k)$ in the Waring--Goldbach problem. We obtain new results for all exponents $k\ge 8$, and in particular establish that $H(k)\le (4k-2)\log k+k-7$ when $k$ is large, giving the first improvement on the classical result of Hua from the 1940s.