Abstract
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 ).
Highlights
A prime gap is the difference between consecutive prime numbers
In this paper we study a further generalization of the prime gap growth problem, viz.: What happens to maximal gaps if we only look at primes in a specific residue class mod q? The new problem statement subsumes, as special cases, maximal prime gaps (k = 1, q = 2) as well as maximal gaps between prime k-tuples (k ≥ 2, q = 2)
To assemble a complete data set of maximal gaps for a given q, we used all H-allowed residue classes r
Summary
A prime gap is the difference between consecutive prime numbers. The sequence of prime gaps behaves quite erratically (see OEIS A001223 [1]). Wolf [12,13,14] expressed the probable size of maximal gaps G ( x ) in terms of the prime-counting function π ( x ): x. Extending the problem statement to prime k-tuples, Kourbatov [16,17] empirically tested (for x ≤ 1015 , k ≤ 7) the following heuristic formula for the probable size of maximal gaps Gk ( x ). To gaps between primes in a residue class—and test them in computational experiments Another goal is to investigate how many maximal gaps should be expected between primes p ≤ x in a residue class, with an additional (optional) condition that p starts a prime constellation of a certain type. Expressions like πc ( x ) or log x denote the square of the respective function
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