Predicting maximal gaps in sets of primes

Abstract

Let $q>r\\ge1$ be coprime integers. Let ${\\mathbb P}_c={\\mathbb P}_c(q,r,{\\cal H})$ be an increasing sequence of primes $p$ satisfying two conditions: (i) $p\\equiv r$ (mod $q$) and (ii) $p$ starts a prime $k$-tuple with a given pattern ${\\cal H}$. Let $\\pi_c(x)$ be the number of primes in ${\\mathbb P}_c$ not exceeding $x$. We heuristically derive formulas predicting the growth trend of the maximal gap $G_c(x)=\\max_{p \\le x}(p -p)$ between successive primes $p,p \\in{\\mathbb P}_c$. Extensive computations for primes up to $10^{14}$ show that a simple trend formula $$G_c(x) \\sim {x\\over\\pi_c(x)}\\cdot(\\log \\pi_c(x) + O_k(1))$$ works well for maximal gaps between initial primes of $k$-tuples with $k\\ge2$ (e.g., twin primes, prime triplets, etc.) in residue class $r$ (mod $q$). For $k=1$, however, a more sophisticated formula $$G_c(x) \\sim {x\\over\\pi_c(x)}\\cdot\\big(\\log{\\pi_c^2(x)\\over x}+O(\\log q)\\big)$$ 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 Cramer s conjecture. We also conjecture that the number of maximal gaps between primes in ${\\mathbb P}_c$ below $x$ is $O_k(\\log x)$.