Like many mathematicians I first came into contact with number theory through Davenport's book The Higher Arithmetic. It was difficult not to be struck by his command of the subject and wonderful expository style. This basic textbook is now into its seventh edition, whilst at a more advanced level, a third edition of Davenport's Multiplicative Number Theory has recently appeared. It is fair to say therefore that Davenport still holds considerable appeal to mathematicians worldwide. On discovering that Davenport had also produced a rather less well-known set of lecture notes treating an area of substantial current interest, I was immediately compelled to try and get it back into print. In doing so, I have tried to preserve in its original format as much of the material as possible, and have merely removed errors that I encountered along the way.

As the title indicates, this book is concerned with the use of analytic methods in the study of integer solutions to certain polynomial equations and inequalities. It is based on lectures that Davenport gave at the University of Michigan in the early 1960s. This analytic method is usually referred to as the ‘Hardy–Littlewood circle method’, and its power is readily demonstrated by the diverse range of number theoretic problems that can be tackled by it. The first half of the book is taken up with a discussion of the method in its most classical setting: Waring's problem and the representation of integers by diagonal forms.