On the distribution of maximal gaps between primes in residue classes

TL;DR

This paper empirically investigates the maximal gaps between primes in specific residue classes, suggesting they are generally bounded by a conjectured trend and follow a Gumbel distribution, with open questions about their limiting behavior.

Contribution

It provides extensive computational evidence for a conjectured upper bound on maximal prime gaps in residue classes and explores their statistical distribution and potential generalizations of classical conjectures.

Findings

Maximal gaps are usually below the predicted trend curve.

Rescaled gaps follow a Gumbel distribution.

Open questions remain about the existence of a limiting distribution.

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.