On the first occurrences of gaps between primes in a residue class

Abstract

We study the first occurrences of gaps between primes in the arithmetic progression (P): $r$, $r+q$, $r+2q$, $r+3q,\dots ,$ where $q$ and $r$ are coprime integers, $q>r\ge 1$. The growth trend and distribution of the first-occurrence gap sizes are similar to those of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value distribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are $O({\log }^{2}x)$ first-occurrence gaps between primes in (P) below $x$, while the number of maximal gaps is only $O(\log x)$. We explore the connection between the asymptotic density of gaps of a given size and the corresponding generalization of Brun's constant. For the first occurrence of gap $d$ in (P), we expect the end-of-gap prime $p\asymp \sqrt{d}\exp (\sqrt{d/\phi (q)})$ infinitely often. Finally, we study the gap size as a function of its index in the sequence of first-occurrence gaps.

24 pages, 4 figures. Sequel to arXiv:1901.03785