Weyl’s inequality

Abstract

The purpose of this chapter is to develop some analytical tools that will be needed to prove the Hardy—Littlewood asymptotic formula for Waring’s problem and other results in additive number theory. The most important of these tools are two inequalities for exponential sums, Weyl’s inequality and Hua’s lemma. We shall also introduce partial summation, infinite products, and Euler products.

The analytic method of Hardy and Littlewood (sometimes called the ‘circle method’) was developed for the treatment of additive problems in the theory of numbers. These are problems which concern the representation of a large number as a sum of numbers of some specified type. The number of summands may be either fixed or unrestricted; in the latter case we speak of partition problems. The most famous additive problem is Waring’s Problem, where the specified numbers are kth powers.... The most important single tool for the investigation of Waring’s Problem, and indeed many other problems in the analytic theory of numbers, is Weyl’s inequality.

H. Davenport [18]