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