On the nth record gap between primes in an arithmetic progression

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.