On the Erdős primitive set conjecture in function fields

Abstract

Erdős proved that $\mathcal{F}(A):={\sum }_{a\in A}\frac{1}{a\log a}$ converges for any primitive set of integers $A$ and later conjectured this sum is maximized when $A$ is the set of primes. Banks and Martin further conjectured that $\mathcal{F}({\mathcal{P}}_1)>\dots >\mathcal{F}({\mathcal{P}}_k)>\mathcal{F}({\mathcal{P}}_{k+1})>\dots$, where ${\mathcal{P}}_j$ is the set of integers with $j$ prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function field ${\mathbb{F}}_q[x]$, investigating the sum $\mathcal{F}(A):={\sum }_{f\in A}\frac{1}{\text{deg}f\cdot q^{\text{deg}f}}$. We establish a uniform bound for $\mathcal{F}(A)$ over all primitive sets of polynomials $A\subset {\mathbb{F}}_q[x]$ and conjecture that it is maximized by the set of monic irreducible polynomials. We find that the analogue of the Banks-Martin conjecture is false for $q=2,3$, and $4$, but we find computational evidence that it holds for $q>4$.

20 pages, 1 table