The Erdos conjecture for primitive sets

TL;DR

This paper investigates the Erdos conjecture concerning primitive sets of integers, exploring bounds on their harmonic sums and connections to prime number races, advancing understanding of prime distribution and primitive sets.

Contribution

It advances the understanding of the Erdos conjecture by establishing new bounds and linking primitive sets to prime number race phenomena.

Findings

Bounded the sum of 1/(a log a) over primitive sets.

Identified connections between primitive sets and prime number races.

Provided partial progress towards the Erdos conjecture.

Abstract

A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on several fronts, and show a connection to certain prime number "races" such as the race between $\pi(x)$ and li$(x)$.