When I began actively pursuing the application of approximate functional equations to number theory, in the early seventies, results of the Ulam–Hyers type were sparse. Moreover, they did not lend themselves to the problems which I had to hand.

It should be emphasised that the method of the stable dual is not concerned with the approximate functional equations that arise, for example, in the theory of the Riemann zeta function. In that theory approximate functional equations are established for certain given functions, mainly sums of exponentials. In a sense an analytic reciprocity law is derived. In the method of the stable dual an unknown function is assumed to satisfy a weak global constraint, and as far as possible the local nature of the function is then determined.

As applied to number theory the method of the stable dual typically gives rise to a complicated approximate functional equation involving several functions and many variables. The first step is to tease out an approximate equation of a more manageable type. This step depends upon the number theoretic and distributional properties of the objects under consideration. The appropriate notion of stability is then determined by the number theoretic application in view. My aim was usually towards an equation with continuous rather than discrete variables. Although by 1980 I had developed a tolerable technique for treating approximate functional equations arising in the study of arithmetic functions, I felt the need to better understand some of the arguments.

The notion of duality and its action in analytic number theory informs this entire work. Emphasis is given to the interplay between the arithmetic and analytic meaning of inequalities. The following remarks place ideas employed in the present work within a broader framework.

1. Conies. By duality the notion of a point conic gives rise to the notion of a line conic. The members of the line conic comprise the tangents to the point conic. Slightly surrealistically we may regard a conic to be a geometric object, defined from the inside by a point locus, and from the outside by a line envelope.

2. Dual spaces. Let V be a finite dimensional vector space over a field F. The dual of V is the vector space of linear maps of V into F. The space V and its dual, V′, are isomorphic.

To every linear map T: V → W between spaces, there corresponds a dual map T′: W′ → V′. In standard notation, the action f(x) of a function f upon x is written 〈x, f〉. The dual map T′ is defined by (Tx, y′) = 〈x, T′y′〉 where x, y′ denote typical elements of V, W′ respectively.

Let V = Fn, W = Fm. We may identify W′ with the set of maps W → F given by k ↦ kty′, where y′ is a vector in W, t denotes transposition.

In this volume I give a unified account of a method in the analytic theory of numbers: the method of the stable dual. The method is particularly effective in the study of arithmetic functions possessing algebraic structure.

The reader may check details of selected applications of the method; turn to the continuing developments discussed in the penultimate chapter; press on to the new. However, works in analysis, especially in analytic number theory, can seem formless. The leading thread, as Hadamard would have called it, [91] p. 105, becomes obscured by a mass of detail. All too often the conclusion will appear atop a pyramid of small steps, each step apparently insignificant. Sometimes this is due to the nature of the subject; sometimes it is not.

As footnote 2 on page 136 of his book [114], Lakatos makes the following remark:

Experience has convinced me of the validity of this statement. Analytic number theory is a dynamic subject, with partial results as moments of clarity. There rarely comes a static final result. Indeed, analytic number theory is largely concerned with method. An heterogeneous nature, allowing opportunities to exhibit frailties of human psychology, gives it a forbidding aspect.