I've found this gem when I was looking for estimates on $\pi(x)$. In the link, the author proves that if you have a balanced multi-set (sum of the inverse of the terms is zero) and a few other conditions, then you get upper and lower bounds to $\psi(x)$. I don't completly understand it yet, but I want to know about ways to work with these bounds.
What I want to know is:
1) Is there a way to search these multi-sets in a way to get closer to 1?
2) What are the methods to find a $x_0$ from which the bounds are valid for all $x \ge x_0$?(Without getting into Dussart's inequalities and Rosser&Schoenfeld method's)