mersenneforum.org [Not a] Disproof of the Riemann hypothesis
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 2022-10-14, 02:50 #12 CuriousKit   "J. Gareth Moreton" Feb 2015 Nomadic 3×5×7 Posts If I'm right in thinking, a straight up disproof requires a counterexample... that is, a non-trivial root where Re(z) ≠ ½. A single line with this value of z will blow everything out of the water. The disproof is interesting in asserting that there are infinitely many zeroes off the critical line, and unfortunately I don't know enough about numbrer theory yet to properly evaluate the proof, but if it is sound, then hopefully a counterexample can be found. Granted, the first trillion or so zeroes have been found, and all lie on the critical line, so I wish you luck!
2022-10-14, 03:23   #13
charybdis

Apr 2020

2·463 Posts

Quote:
 Originally Posted by CuriousKit If I'm right in thinking, a straight up disproof requires a counterexample... that is, a non-trivial root where Re(z) ≠ ½.
No, showing that a counterexample exists is enough to disprove the hypothesis, there's no need to actually find one. We don't know any explicit value x for which pi(x) > li(x), for example.

 2022-10-14, 14:10 #14 Dr Sardonicus     Feb 2017 Nowhere 24·389 Posts It is refreshing to see an attempted proof of a famous unsolved problem in which the author uses a logically sound approach: If the supremum of real parts of zeroes in the critical strip is greater than 1/2, then RH is false. The argument appears to use standard notation and methods, and the paper has references to the literature. These practices will serve the author well in any future submissions. I suspect there to be one or more basic errors in the argument, but no specific error jumped off the page. But I didn't look too hard. It's been too long since I have dealt with this sort of argument. Assuming the author's submission is to a peer-reviewed journal, it will be a referee's task to point out any fatal errors in the argument.
2022-10-14, 22:43   #15
Tatenda

Oct 2022

13 Posts

Quote:
 Originally Posted by Tatenda Thanks for your comment. I have indeed submitted the paper to some journal. However, it would still be good to get some constructive feedback from here.
Quote:
 Originally Posted by Tatenda Dear number theorists, Attached is a possible disproof of the Riemann hypothesis, which you can also download via this link: https://figshare.com/articles/prepri...ction/21261969 So far, I have shown the paper to several non-number theorists, and their general opinion is the approach is interesting and the argument seems to be sound. Your constructive comments are most welcome. Sincerely, Tatenda.
I really appreciate everyone who has shown a genuine interest in my paper. Attached is another approach I devised today. The second approach is shorter and much less technical. As always, your constructive thoughts are most welcome.

Sincerely,

Tatenda.
Attached Files
 Disproof of the Riemann hypothesis.pdf (274.9 KB, 173 views)

 2022-10-14, 23:19 #16 Tatenda   Oct 2022 11012 Posts One might ask why I devised another approach. Well, I was actually investigating whether the first approach can be generalised to all Dirichlet L-functions, and in the process, I stumbled upon the latest approach. Unfortunately, I don't see how either approach extends to all L-functions. Last fiddled with by Tatenda on 2022-10-15 at 00:15
2022-10-15, 01:12   #17
charybdis

Apr 2020

2·463 Posts

Quote:
 Originally Posted by Tatenda I really appreciate everyone who has shown a genuine interest in my paper. Attached is another approach I devised today. The second approach is shorter and much less technical. As always, your constructive thoughts are most welcome. Sincerely, Tatenda.
Check your calculations. The derivative of h(s) doesn't have a pole at s=1.

The big warning sign for me was that you didn't use anything about ψ other than the asymptotics of its error term and the fact that it's 0 for x<2. In place of ψ, I could plug in the function f(x) where f(x) = x for x>=2 and f(x) = 0 for x<2, and your "proof" would show that this function can't exist.

 2022-10-15, 04:27 #18 Tatenda   Oct 2022 13 Posts My bad ! I had made an algebraic mistake on calculating the derivative. Indeed, h'(s) is actually an entire function. You are right, thus the second approach has a crucial flaw. I should have known better concerning how i didn't make use of the deeper properties of psi. Thanks. Last fiddled with by Tatenda on 2022-10-15 at 05:02
2022-10-16, 00:12   #19
Dr Sardonicus

Feb 2017
Nowhere

24·389 Posts

Quote:
 Originally Posted by Tatenda Thanks for your comment. I have indeed submitted the paper to some journal. However, it would still be good to get some constructive feedback from here.
In view of recent developments, you might want to withdraw your submission.

 2022-10-18, 17:01 #20 CuriousKit   "J. Gareth Moreton" Feb 2015 Nomadic 1518 Posts I don't yet know enough about number theory (one day I might!) to say if this is sound or not, but if you can adapt your first approach to tie in ψ more closely, then it might still hold (or instead show the function can exist, in which case it will fall apart).
2022-10-22, 14:33   #21
Dr Sardonicus

Feb 2017
Nowhere

24×389 Posts

My apologies for being so tardy in posting this.

I found links for two different papers by the same author, both purporting to prove the Riemann Hypothesis.

Demonstration of the Riemann Hypothesis
Quote:
 Authors: Tatenda Kubalalika We demonstrate that the Riemann zeta function zeta(s) has no zeros for Re(s)>1/2 (the Riemann Hypothesis). Comments: 4 Pages. Download: PDF Submission history [v1] 2018-11-24 01:45:30 [v2] 2018-11-25 12:59:55
Quote:
 Error Failed to load PDF document.
Apparently the author yanked it.

Then did it again, this time posting to arXiv, claiming it had been submitted to Research in Number Theory.

On the prime zeta function and the Riemann hypothesis
Quote:
 Prime numbers and the Riemann hypothesis Tatenda Kubalalika By considering the prime zeta function, we demonstrate in this note that the Riemann zeta function zeta(s) does not vanish for Re(s)>1/2, which proves the Riemann hypothesis. Comments: 5 pages. Submitted for peer review to the Research in Number Theory Submission history From: Tatenda Kubalalika Isaac [view email] [v1] Mon, 7 Oct 2019 20:02:54 UTC (158 KB) [v2] Mon, 14 Oct 2019 08:21:41 UTC (5 KB) [v3] Fri, 1 Nov 2019 15:25:59 UTC (5 KB) [v4] Wed, 6 Nov 2019 14:05:39 UTC (4 KB) [v5] Tue, 12 Nov 2019 14:32:46 UTC (5 KB) [v6] Thu, 2 Jul 2020 21:53:07 UTC (0 KB) [v7] Thu, 25 Feb 2021 16:28:22 UTC (0 KB)
Clicking on the last two (0 KB) versions, we find an admission by the author that the proof of his Theorem 3 was "fundamentally flawed." Apparently the author was unable to delete the first five versions.

So, there you have it! The OP has both proved and disproved the Riemann Hypothesis!

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