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.