A proof of the Erdős primitive set conjecture

Abstract

A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series f (A) = a∈A 1/(a log a) is uniformly bounded over all choices of primitive sets A. In 1988 he asked if this bound is attained for the set of prime numbers. In this article we answer in the affirmative.As further applications of the method, we make progress towards a question of Erdős, Sárközy, and Szemerédi from 1968. We also refine the classical Davenport-Erdős theorem on infinite divisibility chains, and extend a result of Erdős, Sárközy, and Szemerédi from 1966.Conjecture 1.1 (Erdős primitive set conjecture). For any primitive set A, f (A) ≤ f (P).