Can you share the links to those papers which you claim to be logically "similar" to this one?
And, if you want to be critical, just point out some part(s) of the proof you believe to be flawed and justify why you think so. That's how mathematics works.
If you've studied real analysis some of the integrals don't converge the way the paper claims they do (I think Lubos makes similar comments on the page you linked in one of the sister comments).
May you explicitly and clearly point out the integrals you are referring to ?
Anyway, by quoting Lubos Motl's blog post of 2018/19, i'm lead to guess you are talking about the integrals on the extreme right-hand side of (13) and (14). Indeed, the author clearly and rigorously showed that those integrals are
absolutely convergent and
real-analytic for sigma >
Theta, such that f(sigma) and
g(sigma) both have real-
analytic continuations there.
In the 2018 version that was reviewed by Lubos, the author had an equation of the form
A(s) = B(s)
for Re(s)>1, where A(s) is some improper integral and B(s) is some function that is analytic in some (larger) half-plane. From this, one can only deduce that A(s) has an analytic continuation to the larger plane, but one cannot deduce (as the author did in that 2018 version) that A(s) converges there.
For example,
let A(s) be the integral of x^{-s} w.r.t. x on [1, infty). Then B(s) = (s-1)^{-1} for Re(s) > 1. Notice that B(s) hence A(s) has a meromorphic continuation to the entire complex plane with a (simple) pole only at s=1. However, one cannot deduce from this that A(s) converges for some s with Re(s) < 1. The mistake in the author's previous version is analogous to saying that A(s) converges for Re(s) < 1. However, to his credit, he seems to have been working hard to eliminate these kinds of mistakes.
I guess that's what he means on his arXiv page when he said, "though all of my previous proposed (dis)proofs were flawed, i think the flaws pointed me towards the right direction".
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.
And, if you want to be critical, just point out some part(s) of the proof you believe to be flawed and justify why you think so. That's how mathematics works.