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

Alexei Kourbatov


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.

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International Journal of Statistics and Probability   ISSN 1927-7032(Print)   ISSN 1927-7040(Online)

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