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

Abstract

In the framework of Cramér'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 Cramér'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 Cramér's model.

Predicting Maximal Gaps in Sets of Primes

“…Among two-parameter distributions, the Gumbel extreme value distribution is a very good fit; cf. [46,47]. This was true in all our computational experiments.…”

“…Case k = 1. For maximal gaps G q,r between primes p ≡ r (mod q), the trend function T is given by Equations ( 33), (45) and (46). The rescaling operation has the form…”

On the first occurrences of gaps between primes in a residue class

“…Indeed, we found that the baseline trend T 0 (2, x), even without an error term, satisfactorily describes the most probable sizes of maximal prime gaps G(x). (Another reasonably good choice of the error term in (10) for q = 2 is E m (2, x) = ca(q, x) with a constant c ∈ [0, 1 2 ].) For q > 2, we found that the error term E m (q, x) is positive, and it is best to determine the parameters b 1 and b 2 in (11) a posteriori.…”

“…This invites the question (cf. [10], [13]): do first-occurrence gap sizes, after an appropriate rescaling transformation, obey the same distribution as maximal gap sizes -the Gumbel extreme value distribution? Let d = d q,r (x) denote the size of the first-occurrence gap between primes p ∈ [1, x]∩(P); specifically, the gap d q,r (x) refers to the first occurrence closest to x on the interval [1, x].…”

Verification of the Firoozbakht conjecture for primes up to four quintillion

“…Indeed, the Firoozbakht upper bound (last column) is below the Cramér upper bound by approximately log p k . In Cramér's probabilistic model of primes [1,3] the parameters of the distribution of maximal prime gaps suggest that inequalities (1) and (2) are both true with probability 1; that is, almost all 1 maximal prime gaps in Cramér's model satisfy (1) and (2). One may take this as an indication that any violations of (1) and (2) occur exceedingly rarely (if at all).…”