Can someone explain the decimal constants that are used throughout the proof? For example, on page 52. It's rare to see these kinds of numbers used in mathematical proofs, but I'm sure they were chosen for good reasons.
Unfortunately, nobody can explain anything like this right now. The paper was posted today, is 111 pages long, and it will likely take even professional mathematicians around a year to understand / check it completely.
Often in math papers there is a series of constants appearing in estimates, each of which depends, sometimes in nonobvious ways, on those appearing earlier. Some authors like to give these constants genuine numerical values in order to help keep track of the dependencies between them and, in particular, to make sure that they aren't accidentally treating two different C's as the same. In delicate analytic proofs this sort of thing is particularly important - subtle missed dependencies are one of the places where serious proof attempts can go wrong in hard to find ways. I'm not saying this is what Zhang is doing, but it's one possibility.
I'm not an analytic number theorist, but my understanding is that the proof requires some values to be 0.5 + some positive epsilon to be correct, and for the sake of clarity, these somewhat arbitrary constants are chosen to get to the result.
