Large prime gaps and probabilistic models

2019/08/22 by William D. Banks, Banks, William, Terence Tao +2
Mathematics · #11B83 #11N05 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1908.08613

openalex publication_date 2019/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Beyond pair correlation
• The Difference Between Consecutive Primes
• The large sieve

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• Heuristic Bounded Prime Gaps via a Chaotic Multidimensional Sieve and Random Matrix Theory
• Distribution of Large Gaps Between Primes
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