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A set is primitive if no element of the set divides another. We considerprimitive sets of monic polynomials over a finite field and find naturalgeneralizations of many of the results known for primitive sets of integers. Inparticular we generalize a result of Besicovitch to show that there existprimitive 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 asum considered by Erd\H{o}s for primitive subsets of the integers, and showthat it is uniformly bounded over all primitive sets $A$. We end with ageneralization of work of Martin and Pomerance on the asymptotic growth rate ofthe counting function of a primitive set. Along the way we prove a quantitativeanalogue of the Hardy-Ramanujan theorem for function fields, as well as boundson the size of the $k$-th irreducible polynomial.

On the size of primitive sets in function fields