Prime gaps in residue classes - where CSG > 1 is possible

[mart_r]

My monthly tribute to the world of number theory.
Largest CSG found during the past 4.3 weeks: 1.264846947 (p=113,109,089 / q=3,745,830)

                  max. p searched:
       q <= 1000: 1.908e+13
1000 < q <= 2690: 1.023e+13
2690 < q <= 1e+5: 1.770e+11
1e+5 < q <= 2e+5: 5.720e+10
2e+5 < q <= 5e+5: 4.600e+10
5e+5 < q <= 1e+6: 2.400e+10
1e+6 < q <= 2e+6: 1.20e+10
2e+6 < q <= 4e+6: 3.0e+9
 (only even q are examined)

P.S., @ Bobby: comparison between three ways of calculating CSG, in the case of q=3,613,418 / p₁=487,021:
g/[phi(q)*log²(p₂)]: 0.9241119774
my underappreciated formula: 1.0251848498
g/[phi(q)*log²(p₁)]: 2.2178622671
Finding a CSG above 2 by any other measure is, IMHO, impossible. But I have to be careful here since it's an open problem how large that value can actually be. Data might suggest that a global maximum depends on the ratio log(p)/log(q), in the sense that the largest CSG are attained when log(p)/log(q) is just a little above 1. It may be that CSG cannot be larger than, say, 1.2, if log(p)/log(q) is larger than 2 or thereabouts. All very sketchy at the moment, maybe I'll write a paper about it when the pandemic is over...

[Bobby Jacobs]

Originally Posted by mart_r

g/[phi(q)*log²(p₂)]: 0.9241119774
my underappreciated formula: 1.0251848498
g/[phi(q)*log²(p₁)]: 2.2178622671

I am sorry that your formula is underappreciated. I will appreciate it more. I hope more people use it.

[mart_r]

Originally Posted by Bobby Jacobs

I am sorry that your formula is underappreciated. I will appreciate it more. I hope more people use it.

Thanks for your support