Primitive sets with large counting functions

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 $S(x)\gg F(x)$.

7 pages. Revision includes a strengthening of Theorem 1: an upper bound for S(x) of the same order of magnitude as the lower bound is now established