The asymptotic formula in Waring's problem.

Abstract

We reduce the number of variables required to guarantee the validity of the classical asymptotic formula in Waring's problem. The method employed is an extension of techniques of Heath-Brown and Vaughan. Sharp divisor sum estimates of Hall and Tenenbaum are also crucial. We subsequently establish the validity of an asymptotic formula for the number of representations of a number as the sum of 56 sixth powers, 112 seventh powers, 224 eighth powers and 448 ninth powers of positive integers thereby improving upon the previous upper bounds of Heath-Brown. In a second section of the thesis we adumbrate a technique whereby one may improve upon the error terms that arise in such formulae. In particular, we establish results concerning the representation of a number as the sum of eight cubes--where the cubes may be from restricted sets.