The distribution of maximal prime gaps in Cramer's probabilistic model of primes

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 primes. The result can be derived from a general theorem about intervals between discrete random events occurring with slowly varying probability monotonically decreasing to zero. A straightforward generalization extends the Gumbel limit law to maximal gaps between prime constellations in Cramer's model.

12 pages, 2 figures, errata fixed

Cited by in corpus (5)

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• On the distribution of maximal gaps between primes in residue classes
• Upper bounds for prime gaps related to Firoozbakht's conjecture
• On the first occurrences of gaps between primes in a residue class