Papers › On the Erdős primitive set conjecture in function fields
On the Erdős primitive set conjecture in function fields
Andrés Gómez-Colunga, Charlotte Kavaler, Nathan McNew, Mirilla Zhu
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Erd\H{o}s proved that ℱ(A) := ∑_(a ∈A)1/aloga 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 𝔽_q[x], investigating the sum ℱ(A) := ∑_(f ∈A) 1/(deg f ·q^(deg f)). We establish a uniform bound for ℱ(A) over all primitive sets of polynomials A ⊂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.
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