On the distribution of maximal gaps between primes in residue classes

Abstract

Let q > r ≥ 1 be coprime positive integers. We empirically study the maximal gaps G q,r (x) between primes p = qn + r ≤ x, n ∈ N. Extensive computations suggest that almost always G q,r (x) < ϕ(q) log 2 x. More precisely, the vast majority of maximal gaps are near a trend curve T predicted using a generalization of Wolf's conjecture:where b = b(q, x) = O q (1). The distribution of properly rescaled maximal gaps G q,r (x) is close to the Gumbel extreme value distribution. However, the question whether there exists a limiting distribution of G q,r (x) is open. We discuss possible generalizations of Cramér's, Shanks, and Firoozbakht's conjectures to primes in residue classes.

Predicting Maximal Gaps in Sets of Primes

“…where āc and Tc are defined, respectively, by ( 8) and ( 14), then the resulting Gumbel distributions of h-values will typically have scales α a little below 1. In a similar experiment with random gaps, the scale was also close to 1; see [43,Sect. 3.3].…”

“…To assemble a complete data set of maximal gaps for a given q, we used all H-allowed residue classes r (mod q). For additional details of our computational experiments with maximal gaps between primes p = r + nq (i.e., for the case k = 1), see also [43,Sect. 3].…”

“…At the same time, the right-hand side of ( 43) is less than 2 log x. Thus the right-hand sides of (41) as well as (43) overestimate the actual gap counts N q,r (x) in most cases.…”

“…This is asymptotically equivalent to the following semi-empirical formula for the number of maximal prime gaps up to x (i.e., for the special case k = 1, q = 2; see [43,Sect. 3.4; OEIS A005669]):…”