A proof of the Erd\H{o}s primitive set conjecture

Abstract

A set of integers greater than 1 is primitive if no member in the set divides another. Erd\H{o}s 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 Erd\H{o}s, S\'ark\"ozy, and Szemer\'edi from 1968. We also refine the classical Davenport-Erd\H{o}s theorem on infinite divisibility chains, and extend a result of Erd\H{o}s, S\'ark\"ozy, and Szemer\'edi from 1966.