On the First Occurrences of Gaps Between Primes in a Residue Class
Redmond, WA 98052
Faculty of Mathematics and Natural Sciences
Cardinal Stefan Wyszyński University
We study the first occurrences of gaps between primes in the arithmetic progression
(P): r, r + q,
r + 2q,
r + 3q, . . . ,
where q and r are coprime integers, q > r ≥ 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(log2 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
infinitely often. Finally, we study the gap size as a function of its index
in the sequence of first-occurrence gaps.
Full version: pdf,
(Concerned with sequences
Received June 11 2020; revised versions received September 15 2020,
September 19 2020.
Published in Journal of Integer Sequences,
October 15 2020.
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