An explicit version of Bombieri’s log-free density estimate and Sárközy’s theorem for shifted primes

IF 1 3区数学 Q1 MATHEMATICS

Forum Mathematicum Pub Date : 2024-01-10 DOI:10.1515/forum-2023-0091

Jesse Thorner, Asif Zaman

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引用次数: 0

Abstract

We make explicit Bombieri’s refinement of Gallagher’s log-free “large sieve density estimate near σ = 1

{\sigma=1}

” for Dirichlet L-functions. We use this estimate and recent work of Green to prove that if N ≥ 2

{N\geq 2}

is an integer, A ⊆ { 1 , … , N }

{A\subseteq\{1,\ldots,N\}}

, and for all primes p no two elements in A differ by p - 1

{p-1}

, then | A | ≪ N 1 - 10 - 18

{|A|\ll N^{1-10^{-18}}}

. This strengthens a theorem of Sárközy.

查看原文本刊更多论文

关于移位素数的邦比里无对数密度估计和萨尔科齐定理的明确版本

我们明确了 Bombieri 对 Gallagher 的无对数 "σ = 1 {\sigma=1} 附近的大筛密度估计 "的改进。的 "大筛密度估计"。我们利用这一估计和格林的最新研究成果证明，如果 N ≥ 2 {N\geq 2} 是整数，则 A ⊆ { 1 , ... , N } {A\subseteq\{1,\ldots,N\}} 对于所有素数 p，A 中没有两个元素相差 p - 1 {p-1} ，那么 | A | ≪ N 1 - 10 - 18 {|A|\ll N^{1-10^{-18}}} 。这加强了萨尔科齐的一个定理。

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来源期刊

Forum Mathematicum 数学-数学

CiteScore

1.60

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.