# Primitive sets with large counting functions - Mathematics > Number Theory

Abstract: A set of positive integers is said to be primitive if no element of the setis a multiple of another. If $S$ is a primitive set and $Sx$ is the number ofelements of $S$ not exceeding $x$, then a result of Erd\H os implies that$\int 2^\infty St-t^2\log t dt$ converges. We establish an approximateconverse to this theorem, showing that if $F$ satisfies some mild conditionsand $\int 2^\infty Ft-t^2\log t dt$ converges, then there exists aprimitive set $S$ with $Sx \gg Fx$.

Author: Greg Martin, Carl Pomerance

Source: https://arxiv.org/