In this book I have two aims. My first is to give a coherent account of a general method in analytic number theory, and to develop that method sufficiently far that it solves problems otherwise beyond reach. The method applies the simplest notions from functional analysis, and has its roots in geometry.

My second aim, bound to the first, and to me of equal interest, is a light discussion of the creation of the method as a raising of the underlying philosophical motivation into consciousness. In particular, this offers a paradigm for the application of the method itself.

I wrote the present work and my memoir: The Correlation of Multiplicative and the Sum of Additive Arithmetic Functions together. To facilitate a bridge between the two works I have elaborated the treatment of approximate functional equations given in Chapters 2 and 3 of the monograph. In particular, I preserve the same notation. For permission to do this I thank both the American Mathematical Society and Cambridge University Press.

The memoir applies the method to a problem not treated in this book. Background details in the construction of the method are omitted. Consideration of the problem to hand remains paramount. A large number of auxiliary results are required.

The present work is quite different in nature. The method itself is the object of study. Essential inequalities are derived in detail.