2020/07/05 by Andrés Gómez-Colunga, Charlotte Kavaler, Gómez-Colunga, Andrés +5
Computer Science · Mathematics · #11B83 #11N25 #11T06 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2007.02301
openalex publication_date 2020/07/05 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
Erdős proved that F(A) := ∑a ∈ A(1)/(alog 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 F(P1) > … > F(Pk) > F(Pk+1) > …, where Pj 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 \mathbbFq[x], investigating the sum F(A) := ∑f ∈ A \frac1deg f ⋅ qdeg f. We establish a uniform bound for F(A) over all primitive sets of polynomials A ⊂ \mathbbFq[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.