On the Waring-Goldbach problem for seventh 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 of seven and higher, providing new explicit bounds for large exponents.

Contribution

It introduces improved bounds for the function H(k) in the Waring-Goldbach problem for all k ≥ 7, utilizing recent progress in Vinogradov's mean value theorem.

Findings

Established new bounds for H(k) for all k ≥ 7

Derived explicit bounds for large k

Applied recent theoretical progress to classical problems

Abstract

We apply recent progress on Vinogradov's mean value theorem to improve bounds for the function $H(k)$ in the Waring-Goldbach problem. We obtain new results for all exponents $k \ge 7$, and in particular establish that for large $k$ one has \[H(k)\le (4k-2)\log k-(2\log 2-1)k-3.\]