On the size of primitive sets in function fields

Abstract

A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular we generalize a result of Besicovitch to show that there exist primitive sets in $\mathbb{F}_q[x]$ with upper density arbitrarily close to $\frac{q - 1}{q}$. Then, for a primitive set $A$, we consider the sum $\sum_{a \in A} \frac{1}{q^{\deg a}\deg a}$, the natural analogue in this setting of a sum considered by Erd\H{o}s for primitive subsets of the integers, and show that it is uniformly bounded over all primitive sets $A$. We end with a generalization of work of Martin and Pomerance on the asymptotic growth rate of the counting function of a primitive set. Along the way we prove a quantitative analogue of the Hardy-Ramanujan theorem for function fields, as well as bounds on the size of the $k$-th irreducible polynomial.