Introduction

Abstract

Abstract This book is written from the standpoint of analytic number theory, a subject whose name may seem contradictory. Number theory deals with the behaviour of the integers under addition and multiplication, and analysis with continuity and limit operations. The logicians have told us that any problem in mathematics can be disguised as finding integer solutions of a set of simultaneous quadratic equations. This means that we must be prepared to use any method that comes to hand. We may classify problems in number theory only by the methods used, not by the form in which they are stated. An argument is algebraic if it uses only addition and multiplication, and analytic if it uses the ordering of the integers, or the ordering of the rational numbers within the real line, which is a consequence of the ordering of the integers. From this point of view ‘elementary number theory’ is not a third category, but a description of the early levels. It is in this sense that the first part of this book is ‘elementary’.