Riemann Hypothesis Disproved?
figshare.com
figshare.com
I'm not even going to bother opening the link.
Sometimes it is much easier to prove a counterexample exists than to demonstrate that a specific number is a counterexample.
Here's the canonical example of this:
We want to show that there exists irrational numbers a, b s.t. a^b is rational (with b != 0).
If sqrt(2) ^ sqrt(2) is rational then we are done. Otherwise, (sqrt(2) ^ sqrt(2)) ^ sqrt(2) = 2 so that is a counterexample.
In this case, it is possible to prove that sqrt(2) ^ sqrt(2) is irrational, but this requires much more work.
Whether you want to put a lot of effort into it is your call but it's still interesting to see what kind of approach they chose.
He's some guy with no credentials, graduated as an accountant. Clearly has a knack for math and is not terrible at it (mathoverflow), but otherwise has no social credit supporting his credibility (twitter, LinkedIn).
Doesn't mean he's necessarily wrong, maybe he got it. It's not impossible that he's some kind of contemporary Ramanujan, but yeah.. unlikely.
[1] https://web.archive.org/web/20220128195023/https://motls.blo...
[2] https://en.wikipedia.org/wiki/Lubo%C5%A1_Motl
[3] https://www.scienceforums.net/topic/124368-has-the-riemann-h...
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."
Note that it's not a "per review paper", it's just a PFD shared somewhere, a "preprint" if you like a formal word.
It looks short (4 pages) and looks quite readable, so I guess that if you wait a week or two it will be a major new IF true.
My guess is it's false. It looks like the author made a similar claim two years ago and it's now fixing some details. With a loooooooooooooooooooong proof that is possible because only the author can understood it good enough to fix it, but with a proof that is so short a minor error would have been fixed soon.
http://math.ahu.edu.cn/_upload/article/files/85/03/f191ded54...
[1] https://web.archive.org/web/20220128195023/https://motls.blo...
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..."
Being such a short paper if people in this field thought it was close to actually disproving the Riemann Hypothesis likely another mathematician could takes the proofs concept, fix it, publish it, and have it peer reviewed to snipe the claim of disproving the Riemann Hypothesis.
This is exactly what happened with Perelman's proof of the Poincare conjecture: https://www.newyorker.com/magazine/2006/08/28/manifold-desti...
But yes, if the proof is only missing easy plug-able gaps then likely a mathematician would reach out to the author so they could co-author the paper. Anything else would be rude.
I was more meaning in the case where the proof has some insight but not apply it in a way that actually disproves the Riemann Hypothesis.
A quick Google search of the author would also show that they have published several similar proofs before [0] which were also wrong.
..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.
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).
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.
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.
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.
Meme: an idea, behavior, style, or usage that spreads from person to person within a culture