On the Erdős primitive set conjecture in function fields

Gómez-Colunga, Andrés; Kavaler, Charlotte; McNew, Nathan; Zhu, Mirilla

doi:10.1016/j.jnt.2020.09.001

Cited by 2 publications

(3 citation statements)

References 10 publications

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“…We have already described a few in the introduction, including Conjecture 1.4, as well as whether p = 2 is Erdős strong. We also note recent work has studied variants of the problem in function fields F q [x], see [6], [7]. In addition, it would be interesting to further extend the classical study of…”

Section: Closing Remarksmentioning

confidence: 93%

“…Sets of L-multiples play a central role in our proof of Theorem 1.2, as the mathematical structures arising from a probabilistic interpretation of (1.4), 6 and implicit in the original 1935 argument of Erdős [13]. 7 As such it is natural to pose the L-primitive analogue of Conjecture 1.1, namely that f (A) ≤ f (P) for all L-primitive sets A.…”

Section: Thus Dividing Bymentioning

confidence: 99%

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A proof of the Erdős primitive set conjecture

Lichtman¹

2022

Preprint

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A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series f (A) = a∈A 1/(a log a) is uniformly bounded over all choices of primitive sets A. In 1988 he asked if this bound is attained for the set of prime numbers. In this article we answer in the affirmative.As further applications of the method, we make progress towards a question of Erdős, Sárközy, and Szemerédi from 1968. We also refine the classical Davenport-Erdős theorem on infinite divisibility chains, and extend a result of Erdős, Sárközy, and Szemerédi from 1966.Conjecture 1.1 (Erdős primitive set conjecture). For any primitive set A, f (A) ≤ f (P).

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Section: Closing Remarksmentioning

confidence: 93%

Section: Thus Dividing Bymentioning

confidence: 99%

A proof of the Erdős primitive set conjecture

Lichtman¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…We have already described a few in the introduction, including Conjecture 1.4, as well as whether 𝑝 = 2 is Erdős strong. We also note recent work has studied variants of the problem in function fields F 𝑞 [𝑥]; see [6], [7]. In addition, it would be interesting to further extend the classical study of sets of (all) multiples and of primitive sets,for example, see Hall [21] or Halberstam-Roth [20, §5], to sets of L-multiples and L-primitive sets.…”

Section: Closing Remarksmentioning

confidence: 98%

A proof of the Erdős primitive set conjecture

Lichtman

2023

Forum of Mathematics, Pi

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A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \sum _{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets A. In 1986, he asked if this bound is attained for the set of prime numbers. In this article, we answer in the affirmative. As further applications of the method, we make progress towards a question of Erdős, Sárközy and Szemerédi from 1968. We also refine the classical Davenport–Erdős theorem on infinite divisibility chains, and extend a result of Erdős, Sárközy and Szemerédi from 1966.

show abstract

On the Erdős primitive set conjecture in function fields

Cited by 2 publications

References 10 publications

A proof of the Erdős primitive set conjecture

A proof of the Erdős primitive set conjecture

A proof of the Erdős primitive set conjecture

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