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 $f(A)={\sum }_{a\in A}1/(a\log a)$ is uniformly bounded over all choices of primitive sets $A$. In 1986 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.

22 pages. The author was informed that the Erdős primitive set conjecture appears in print at least since 1974, and was possibly posed much earlier still