On the distribution of maximal gaps between primes in residue classes

Abstract

Let $q>r\ge 1$ 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 $G_{q,r}(x)<\phi (q){\log }^{2}x$. More precisely, the vast majority of maximal gaps are near a trend curve $T$ predicted using a generalization of Wolf's conjecture: $$G_{q,r}(x)\sim T(q,x)=\frac{\phi (q)x}{\mathrm{li}(x)}(2\log \frac{\mathrm{li}(x)}{\phi (q)}-\log x+b),$$ 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.

25 pages, 4 figures, 3 tables