16 karma · joined August 14, 2022
1 \leq x < 2.
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.
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.
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".
Credit where it's due.
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.
..though all of my previous (dis)proofs were flawed, i think the flaws pointed me towards the right direction..."
it seems they are quite confident in the current version.
This is exactly what happened with Perelman's proof of the Poincare conjecture: https://www.newyorker.com/magazine/2006/08/28/manifold-desti...
On their arXiv page: https://arxiv.org/abs/2006.12546, the author states that "though all my previous proposed (dis)proofs were flawed, i think the flaws pointed me towards the right direction..."
On their arXiv page: https://arxiv.org/abs/2006.12546, the author states that "though all my previous proposed (dis)proofs were flawed, i think the flaws pointed me towards the right direction."