Optimal primitive sets with restricted primes

TL;DR

This paper investigates Erdős's conjecture on primitive sets, focusing on subsets of primes and their associated natural numbers, establishing conditions under which these sets are optimal with respect to certain sum inequalities.

Contribution

The paper introduces the concept of Erdős-best sets within restricted prime subsets and proves their optimality under specific reciprocal sum conditions.

Findings

If the reciprocal sum of P is small enough, P is Erdős-best among primitive subsets of N(P).

The set of twin primes exceeding 3 is Erdős-best among the corresponding primitive sets.

For t>1, P is t-best if the sum of p^{-t} over P is sufficiently small.

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).