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

TL;DR

This paper proves Erdős's conjecture that the sum of 1/(a log a) over primes attains the maximum among primitive sets, advancing understanding of primitive sets and related number theory questions.

Contribution

The paper provides a proof that the sum over primes reaches the maximum for primitive sets, confirming Erdős's conjecture and extending classical results in number theory.

Findings

Confirmed Erdős's primitive set conjecture.

Extended Davenport-Erdős theorem on divisibility chains.

Made progress on a 1968 question by Erdős, Sárközy, and Szemerédi.

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.