Further improvements in Waring’s problem. Ill Eighth powers

Abstract

In this paper we continue our development of the methods of Vaughan & Wooley, these being based on the use of exponential sums over integers having only small prime divisors. On this occasion we concentrate on improvements in the estimation of the contribution of the major arcs arising in the efficient differencing process. By considering the underlying diophantine equation, we are able to replace certain smooth Weyl sums by classical Weyl sums, and thus we are able to utilize a number of pruning processes to facilitate our analysis. These methods lead to improvements in Waring’s problem for larger k . In this instance we prove that G (8) ⩽ 42, which is to say that all sufficiently large natural numbers are the sum of at most 42 eighth powers of integers. This improves on the earlier bound G (8) ⩽ 43.