Quote:
Originally Posted by mart_r
It seems I was even more lucky than I originally thought with that gap at q=152. It gets increasingly hard to find gaps even with CSG>0.9.

Specifically, between 5*10
^{12} and 7.3*10
^{12} for q<=1000 the largest CSG is about 0.8849, which puts a large question mark behind ever finding another gap that is larger than both q² and
(q) * log²p. I'm kind of heuristically challenged there.
From what I gather, the conjectures of Granville et al. for gaps between consecutive primes only, and at best, apply to prime gaps in AP with common difference q if the bounding prime p' = p+q*k is no larger than (p+n)² for very small n (n<2, say).
On the other hand, primes in AP may behave more random, as it were, than the primes themselves. Even if the gaps with CSG>1 is finite for any fixed q, is it plausible to expect that CSG can be arbitrarily large, maybe even as a function of q, how do you write it, f(q) > O(1)?
But I won't leave unanswered questions here today without providing a new record gap with both (q, r) prime:
p = 7,302,961,447
q = 214,451
r = 47,093
k = 576
g/
(q)/log²p' = 1.1150385797428...
CSG (via sum Ri') = 1.1158450619299...
Which is also quite remarkable as it's in the upper decile of all gaps with CSG>1.