Sieves of Dimension 1 +

with fκ(s) as large as possible (resp. Fκ(s) as small as possible) given that the above inequality holds for all choices of A satisfying (1). Selberg [2] has shown (in a much more general context) that the functions fκ(s), Fκ(s) are continuous, monotone, and computable for s > 1, that they do not change if we replace (1) with (2), and that they tend to 1 exponentially as s goes to infinity. More specifically, fκ(s) and Fκ(s) can be defined as follows. LetM be the collection of all finite multisubsets of [0, 1], and for S ∈M let Σ(S) be the sum of the elements of S and |S| be the number of elements of S (both counted with multiplicity). When we write sums like ∑ A⊆S , we also count subsets A with multiplicity, so such a sum will always have 2|S| summands. Let λ : M→ R be a piecewise continuous function supported on S with Σ(S) ≤ 1, and define a function θ :M→ R by