Dense admissible sets

Abstract

Call a set of integers {b ₁, b ₂,..., b _k} admissible if for any prime p, at least one congruence class modulo p does not contain any of the b _i. Let ρ ^*(x) be the size of the largest admissible set in [1, x]. The Prime k-tuples Conjecture states that any for any admissible set, there are infinitely many n such that n+b ₁,... n+b ₂, n+b _k are simultaneously prime. In 1974, Hensley and Richards [3] showed that ρ ^*(x)>π(x) for x sufficiently large, which shows that the Prime k-tuples Conjecture is inconsistent with a conjecture of Hardy and Littlewood that for all integers x,y ≥ 2,

$$\pi (x + y) \leqslant \pi (x) + \pi (y).$$

In this paper we examine the behavior of ρ ^*(x), in particular, the point at which ρ ^*(x) first exceeds π(x), and its asymptotic growth.