{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-erdos-primitive-set-conjecture-in","title":"On the Erdős primitive set conjecture in function fields","arxiv_id":"2007.02301","date":"2020-07-05","proceeding":null,"authors":["Andrés Gómez-Colunga","Charlotte Kavaler","Nathan McNew","Mirilla Zhu"],"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 $q > 4$.","url_abs":"https://arxiv.org/abs/2007.02301v1","url_pdf":"https://arxiv.org/pdf/2007.02301v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-erdos-primitive-set-conjecture-in","repo_url":"https://github.com/agreatnate/Erdos-Sum-Function-Fields","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}