As I understood from this discussion, if RH is false, it can be provably false because of Robin-Ramanujan inequalities

@მამუკა ჯიბლაძე it is well known that if there exists an number that do not obey Lagarias inequality, then it must be a colossally abundant number.

Thanks for the explanation @GHfromMO . It's very clear for me now. Also, shouldn't there be $\exp(\gamma)$ in the denominator of the fraction on the RHS ? Your next statement that it tends to zero is still valid as $\exp(\gamma)$ is a constant, but I'm just making sure my understanding is correct as I don't really have a math background.

Thanks @Charles , I understood that they are not claiming RH can be proved by checking a finite number of cases, I wanted to understand how can we reduce K to any 1+ϵ by checking more cases and why can't we go beyond $1$ ? Perhaps, I should edit my question.