On the nth record gap between primes in an arithmetic progression

TL;DR

This paper investigates the behavior of record gaps between primes in arithmetic progressions, proposing conjectures, providing numerical evidence, and analyzing the distribution and growth of these gaps.

Contribution

It introduces new conjectures on the growth and distribution of record prime gaps in arithmetic progressions, supported by numerical data and heuristic arguments.

Findings

Heuristically, the number of record gaps grows like 2 times log x.

Conjecture that record gaps are bounded by a quadratic function of n.

Distribution of gaps is skewed, resembling Gumbel and lognormal distributions, with skewness decreasing as n increases.

Abstract

Let $q>r\ge1$ be coprime integers. Let $R(n,q,r)$ be the $n$th record gap between primes in the arithmetic progression $r$, $r+q$, $r+2q,\ldots,$ and denote by $N_{q,r}(x)$ the number of such records observed below $x$. For $x\to\infty$, 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 $$R(n,q,r)<\varphi(q)n^2+(n+2)q\log^2 q.$$ 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 $q\gtrsim200$ and $n\approx10$, 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.