Optimal primitive sets with restricted primes

Abstract

A set of natural numbers is primitive if no element of the set divides another. Erd\H{o}s conjectured that if S is any primitive set, then \sum_{n\in S} 1/(n log n) \le \sum_{n\in \P} 1/(p log p), where \P denotes the set of primes. In this paper, we make progress towards this conjecture by restricting the setting to smaller sets of primes. Let P denote any subset of \P, and let N(P) denote the set of natural numbers all of whose prime factors are in P. We say that P is Erd\H{o}s-best among primitive subsets of N(P) if the inequality \sum_{n\in S} 1/(n log n) \le \sum_{n\in P} 1/(p log p) holds for every primitive set S contained in N(P). We show that if the sum of the reciprocals of the elements of P is small enough, then P is Erd\H{o}s-best among primitive subsets of N(P). As an application, we prove that the set of twin primes exceeding 3 is Erd\H{o}s-best among the corresponding primitive sets. This problem turns out to be related to a similar problem involving multiplicative weights. For any real number t>1, we say that P is t-best among primitive subsets of N(P) if the inequality \sum_{n\in S} n^{-t} \le \sum_{n\in P} p^{-t} holds for every primitive set S contained in N(P). We show that if the sum on the right-hand side of this inequality is small enough, then P is t-best among primitive subsets of N(P).