[Not a] Disproof of the Riemann hypothesis

[CuriousKit]

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!

[charybdis]

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.

[Dr Sardonicus]

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.

[Tatenda]

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.

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.

[Tatenda]

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.

[charybdis]

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.

[Tatenda]

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.

[Dr Sardonicus]

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.

[CuriousKit]

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).

[Dr Sardonicus]

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

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

Alas, trying to download this paper has the following result:

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

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
<snip>
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!