A proof of the Erdos primitive set conjecture

A set of integers greater than 1 is primitive if no member in the set divides another. Erdos proved in 1935 that the series f (A) = Sigma a is an element of 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 Erdos, Sarkozy and Szemeredi from 1968. We also refine the classical Davenport-Erdos theorem on infinite divisibility chains, and extend a result of Erdos, Sarkozy and Szemeredi from 1966.