On the distribution of maximal gaps between primes in residue classes

Abstract

Let $q>r\ge1$ be coprime positive integers. We empirically study the maximal gaps $G_{q,r}(x)$ between primes $p=qn+r\le x$, $n\in{\mathbb N}$. Extensive computations suggest that almost always $G_{q,r}(x)<\varphi(q)\log^2x$. More precisely, the vast majority of maximal gaps are near a trend curve $T$ predicted using a generalization of Wolf's conjecture: $$G_{q,r}(x) ~\sim~ T(q,x)={\varphi(q)x\over{\rm li}(x)} \Big(2\log{{\rm li}(x)\over\varphi(q)} - \log x + b\Big),$$ 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 Cramer's, Shanks, and Firoozbakht's conjectures to primes in residue classes.