On the nth record gap between primes in an arithmetic progression

Abstract

Let q > r ≥ 1 be coprime integers. Let R(n, q, r) be the nth record gap between primes in the arithmetic progression r, r + q, r + 2q, . . . , and denote by N q,r (x) the number of such records observed below x. For x → ∞, we heuristically argue that if the limit of N q,r (x)/ log x exists, then the limit is 2. We also conjecture that R(n, q, r) = O q (n 2 ). Numerical evidence supports the conjectural a.s. upper bound R(n, q, r) < ϕ(q)n 2 + (n + 2)q log 2 q.The median (over r) of R(n, q, r) grows like a quadratic function of n; so do the mean and quartile points of R(n, q, r). For fixed values of q ≥ 200 and n ≈ 10, the distribution of R(n, q, r) is skewed to the right and close to both Gumbel and lognormal distributions; however, the skewness appears to slowly decrease as n increases. The existence of a limiting distribution of R(n, q, r) is an open question.Mathematics Subject Classification: 11N05, 11N13, 11N56

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“…(50) and (51) is a restatement of Eqs. (40) and (41). It would be logically unsound to suppose that log(P(n) − P(n − 1)) ?…”