On the Erd\H{o}s primitive set conjecture in function fields

TL;DR

This paper explores the Erdős primitive set conjecture within the context of function fields, establishing bounds and examining the maximality of prime sets, with computational evidence for certain finite fields.

Contribution

It extends the Erdős primitive set conjecture to function fields, providing bounds and analyzing the maximality of prime polynomial sets, and refutes a related conjecture for small finite fields.

Findings

Established a uniform bound for primitive sets in function fields.

Conjectured the set of monic irreducible polynomials maximizes the sum.

Disproved the Banks-Martin analogue for small finite fields, with evidence for larger fields.

Abstract

Erd\H{o}s 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 $\mathcal{F}(\mathcal{P}_1) > \ldots > \mathcal{F}(\mathcal{P}_k) > \mathcal{F}(\mathcal{P}_{k+1}) > \ldots$, 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 $q > 4$.