Large prime gaps and probabilistic models

• 成果类型: Article
• 署名作者: Banks, William; Ford, Kevin; Tao, Terence
• 署名单位: University of Missouri System; University of Missouri Columbia; University of Illinois System; University of Illinois Urbana-Champaign; University of California System; University of California Los Angeles
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01199-0
• 发表日期: 2023
• 页码: 1471-1518
• 关键词: tuples; number; SUM

摘要

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval [1, x]. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.