Predicting Maximal Gaps in Sets of Primes

Let q > r ≥ 1 be coprime integers. Let P c = P c ( q , r , H ) be an increasing sequence of primes p satisfying two conditions: (i) p ≡ r (mod q ) and (ii) p starts a prime k -tuple with a given pattern H . Let π c ( x ) be the number of primes in P c not exceeding x . We heuristically derive formulas predicting the growth trend of the maximal gap G c ( x ) = max p ′ ≤ x ( p ′ − p ) between successive primes p , p ′ ∈ P c . Extensive computations for primes up to 10 14 show that a simple trend formula G c ( x ) ∼ x π c ( x ) · ( log π c ( x ) + O k ( 1 ) ) works well for maximal gaps between initial primes of k -tuples with k ≥ 2 (e.g., twin primes, prime triplets, etc.) in residue class r (mod q ). For k = 1 , however, a more sophisticated formula G c ( x ) ∼ x π c ( x ) · log π c 2 ( x ) x + O ( log q ) gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes ( k = 1 , q = 2 , r = 1 ). The distribution of appropriately rescaled maximal gaps G c ( x ) is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cramér’s conjecture. We also conjecture that the number of maximal gaps between primes in P c below x is O k ( log x ) .