Erdős原始集猜想的证明

IF 2.8 1区数学 Q1 MATHEMATICS

Forum of Mathematics Pi Pub Date : 2022-02-04 DOI:10.1017/fmp.2023.16

J. Lichtman

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引用次数: 3

摘要

大于1的整数集合是原始的，如果集合中没有能整除另一个整数的元素。Erdős在1935年证明了级数$f(A) = \sum _{a\in A}1/(a \log a)$在所有原始集合a的选择上是一致有界的。1986年，他问质数集合是否能得到这个界。在本文中，我们的回答是肯定的。作为该方法的进一步应用，我们在求解Erdős、Sárközy和1968年以来的szemersamedi问题方面取得了进展。并对无限可分链上的经典Davenport-Erdős定理进行了改进，推广了Erdős、Sárközy和szemer的结论。

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A proof of the Erdős primitive set conjecture

Abstract 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.

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来源期刊

Forum of Mathematics Pi Mathematics-Statistics and Probability

CiteScore

3.50

自引率

0.00%

发文量

审稿时长

19 weeks

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