Primitive sets with large counting functions

TL;DR

This paper explores the relationship between the growth of primitive sets and their counting functions, establishing conditions under which certain growth patterns imply the existence of primitive sets with those properties.

Contribution

It provides an approximate converse to Erdős's result, linking the convergence of an integral involving a primitive set’s counting function to the existence of such sets with specified growth.

Findings

Established conditions for primitive sets with large counting functions.

Connected integral convergence to the existence of primitive sets with prescribed growth.

Extended understanding of the structure and size of primitive sets.

Abstract

A set of positive integers is said to be primitive if no element of the set is a multiple of another. If $S$ is a primitive set and $S(x)$ is the number of elements of $S$ not exceeding $x$, then a result of Erd\H os implies that $\int_2^\infty (S(t)/t^2\log t) dt$ converges. We establish an approximate converse to this theorem, showing that if $F$ satisfies some mild conditions and $\int_2^\infty (F(t)/t^2\log t) dt$ converges, then there exists a primitive set $S$ with $S(x) \gg F(x)$.