The Distribution of Maximal Prime Gaps in Cramer's Probabilistic Model of Primes

2014/01/31 by Alexei Kourbatov
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics #Discrete mathematics #Extreme value theory #Generalization #Gumbel distribution #History and Theory of Mathematics #Limit (mathematics) #Mathematical analysis #Mathematics #Prime (order theory) #Probabilistic logic #Statistics #math.NT #math.PR #math.ST #msc:11N05 #msc:60G70 #msc:62E20 #stat.TH

paper · pdf · doi:10.5539/ijsp.v3n2p18

published as International Journal of Statistics and Probability; Vol. 3, No. 2, 2014, pp.18-29 · 12 pages, 2 figures, errata fixed

openalex publication_date 2014/03/20 · arxiv created 2014/09/27 · arxiv updated 2014/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In the framework of Cramer's probabilistic model of primes, we explore the exact and asymptotic distributions of maximal prime gaps. We show that the Gumbel extreme value distribution exp(-exp(-x)) is the limit law for maximal gaps between Cramer's random

Citations

• Some heuristics on the gaps between consecutive primes
• Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸
• Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes
• Limiting forms of the frequency distribution of the largest or smallest member of a sample
• On the order of magnitude of the difference between consecutive prime numbers
• Prime Numbers and Computer Methods for Factorization