20221014, 02:50  #12 
"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 nontrivial 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!

20221014, 03:23  #13  
Apr 2020
2·463 Posts 
Quote:


20221014, 14:10  #14 
Feb 2017
Nowhere
2^{4}·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 peerreviewed journal, it will be a referee's task to point out any fatal errors in the argument. 
20221014, 22:43  #15  
Oct 2022
13 Posts 
Quote:
Quote:
Sincerely, Tatenda. 

20221014, 23:19  #16 
Oct 2022
1101_{2} Posts 
One might ask why I devised another approach.
Well, I was actually investigating whether the first approach can be generalised to all Dirichlet Lfunctions, and in the process, I stumbled upon the latest approach. Unfortunately, I don't see how either approach extends to all Lfunctions. Last fiddled with by Tatenda on 20221015 at 00:15 
20221015, 01:12  #17  
Apr 2020
2·463 Posts 
Quote:
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. 

20221015, 04:27  #18 
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 20221015 at 05:02 
20221016, 00:12  #19 
Feb 2017
Nowhere
2^{4}·389 Posts 

20221018, 17:01  #20 
"J. Gareth Moreton"
Feb 2015
Nomadic
151_{8} 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).

20221022, 14:33  #21  
Feb 2017
Nowhere
2^{4}×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:
Quote:
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:
So, there you have it! The OP has both proved and disproved the Riemann Hypothesis! 

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