I was referring to the ones in Motl's blog. I wasn't aware this paper was updated compared to that one to remove those problems. Either way, looking at the paper I see a few things I can't follow. For example for theorem 1 he says for d>0 there exists \psi(y) = some expression uniformly for y≥1. The source he quotes is Montgomery's book on multiplicative number theory. Looking at the theorem in that book (Theorem 6.9) I can't see how what he says follows from it. The book is looking at the distribution of prime numbers with some error bounds. There is nothing in the paper about any sort of error bounds. Quite often such papers when submitted to arXiv get a quick look-over and if the professional mathematicians don't think it's worth much it gets thrown into the "General Mathematics" section, whereas more legitimate attempts would be put in the number theory section.
I really wonder if you have studied basic analytic number theory, because that bound is actually well-known and is equivalent to the PNT. Anyway, Montgomery-Vaughan explicitly and clearly states in equation 6.12 that there exists some constant c>0 such that
psi(x) = x + O(x exp(-c√log x))
uniformly for x \geq 2. Since psi(x)=0 for x in [1, 2), this uniformity trivially extends to x \geq 1.
Also, arXiv moderators don't review papers, they are too busy for that. Submissions claiming to solve famous open problems are classified basing more on author reputation/submission history. I personally know a few moderators in math and physics.
Yes you would be right I haven't taken a class on analytic number theory. Perhaps excuse my lack of knowledge on the subject but why would psi(x)=0 for x in [1, 2)? Λ(2) is not 0. You are right that arXiv moderators don't review papers in the sense of peer review but they still check over them and I haven't heard of an occasion where something that was originally placed in General Mathematics or General Physics was later moved to another category. Safe to say if there was anything substantial in the proof we would have heard of something from the experts in number theory by now.
Anyway, your first comment would make one think that you are an expert in analytic number theory who found some crucial flaw in the proof. But since that's clearly not the case, i won't comment any further on this thread.
