User blog comment:C7X/Supremum Method for Defining an OCF/@comment-35470197-20200114061905/@comment-35470197-20200114102436

Actually better, but the OCF is bounded by the constant ζ_0 and still does not depend on ν. I think that what the OP actualy needs is not an alternative candidate, but is the way to understand the OCF. Then it is good for the OP to read the article on Buchholz's function.