Please explain to me, what do Terence Tao and other mathematicians who write books about the Collatz Conjecture, what do they want to do in general? Do they want to prove this hypothesis, or disprove it?
What can we generally conclude from these books? what are the mathematical predictions about this problem ,what is it in general?
There is The Ultimate Challenge: the 3x+1 Problem by Jeffrey Lagarias (which is a collection of papers by various writers). There are papers and chapters in books by others.
But generally, the books on the Collatz conjecture seem to be the self-published writings of cranks.