Large prime gaps and probabilistic models

William Banks, Kevin Ford, Terence Tao

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1471-1518
Number of pages48
JournalInventiones Mathematicae
Issue number3
StatePublished - Sep 2023

ASJC Scopus subject areas

  • General Mathematics


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