I wrote a paper on this very notion of exponential iterations.. Now it was rejected for no stated reason.. the funny thing is both AMS and the British Mathematical society wrote me rejection emails that are exactly 6 hrs apart and in sequence of British time and American time... This means they collaborated on a response. The proof lies in the fact that an exponentiation in power towers of the zeta function results in convergence too a special negative value, -0.295.... which means the power towers must go through Zeros, and thus the Zeta function must go through a zero. The theorem that clinches it is due to Thron, which states that exponential complex power towers must converge to rational forms s/(1-s) = exp(2*i*theta) = (1/2 + I*tan(theta)/2)/(1/2 - I*tan(theta)/2).... the half line.

Impressive!!

Few mathematicians are convinced by this. About a year ago Atiyah also circulated an alleged short proof of the Feit-Thompson theorem in group theory which came nowhere near to convincing anyone. So it looks as if it's becoming a habit. A sad ending to a great career.

My approach is to raise power towers until one hits a roots, then it converges to a constant, and the condition of convergence is when the s/(1-s) = exp(2*I*theta)....Thron... This is of course the half-line...So an infinite iteration of exponentials (power towers of exponentials) was the paper I submitted to AMS and the British Mathematical Society....Atiyah was an associate editor at the time...just like Atiyah is stating will lead to the solution. When a mathematician says you must collaborate with their colleagues for you to get published, your ears prick up waiting for more insult.

Michael,

I wish you the best in your research endeavors.

You study seems impressive.

And also please be not be the second person who will deny to cash in the one Million prize :)

Emmanouil

Great unsolved problems in mathematics resembles of those billion dollar experiments in physics ,like Higgs boson, GW and so on. As a physicist i can say that those experiments in physics are most likely fakes. Regarding mathematics , I have had an opportunity to ask one of the leading world mathematician about Wiles proof of the last Fermat theorem. That proof by Wiles have more than 100 pages of highly abstract math. So answer to my question , did really Wiles proved the last Fermat theorem was as follows " If somebody wants to spent half a year of reading that proof day by day , doing nothing else and just try follow what Wiles wrote , then may be we get an answer , if the guy reading that did not made mistake by himself." No question Sir Atiyah is a great mathematician as Dr.Higgs is a great physicist. Who has another billion dollars to spent on verification of Higgs boson experiment? No such money, and no humans to read half a year proofs. Let do it AI.

@ Emmanouil Markoulakis

Probably you (and many other people) have already "googled" to learn more about Atiyah's sensational announcement. Being a number theorist, but not an expert on RH, I can only stick to the usual ethical conduct of my profession : wait and see. The official validation or not of a big mathematical result is a very serious painstaking matter, as has been shown by the widely publicized story of Wiles' demonstration of Fermat's Last Theorem (*). Concerning the reactions of the mathematician community, I recommend the two following links (the AMS one gives a few technical details) which seem to me to express what should be the "reasonable doubts" of professional mathematicians (not of journalists nor people outside the field).

https://blogs.ams.org/blogonmathblogs/2018/10/01/on-michael-atiyah-and-the-riemann-hypothesis/

https://aperiodical.com/2018/09/atiyah-riemann-hypothesis-proof-final-thoughts/

(*) To answer some (common sensical but) gratuitous doubts on Wiles' proof, allow me to add the following excerpt from a discussion on the Stack Exchange math. site

According to a report by J-P. Bourguignon, the director of the IHES (the French institute for math. and theoretical physics which, I think, all researchers know of), the total number of mathematicians world-wide at the end of the 20th century was as high as 80,000. And the term "mathematician" was reserved to those having a level of formation equivalent to a PHD, and whose professional activity genuinely makes room to research in math. or assimilation of math. results. So, even with such a restrictive definition, the 0,5% evoked by @Adrien Keister amounts to roughly 400 persons who could have a good enough global (= not necessarily detailed) comprehension of what Wiles' proof is about. Of course the number of experts able to follow and/or use (hence check) Wiles' results should be actually smaller, but not so small as to allow doubts. By the way, your professor's satement: "Andrew Wiles's proof was not actually approved. Only the mathematicians accepted this as true" looks like non sense to me.

More precisely : 1) Outside the usual "popularization" comments, there have been many serious published accounts at post-PHD level, meant to explain the backbone of Wiles' demonstration, by known workers in the field such as Boston, Frey, Rubin, Stein, Stevens, etc. 2) At the research level, many academic sessions or seminars (in particular the reference "Séminaire Bourbaki", with a talk given by J-P. Serre) world-wide have been devoted to Wiles' work 3) Much more important, Wiles' results are not limited to FLT, which in itself is only an isolated riddle (deeply buried, but nonetheless a riddle). Wiles' real achievement was the proof of the Shimura-Taniyama-Weil conjecture (= the "modularity" of elliptic curves defined over Q). His (many) new results have been "tested" in the sense that they paved the way to the proof of Serre's conjecture on "modular representations", from which FLT can now be derived by a half-page long argument ! This, together with STW, is considered as the first step toward the realization of the Langlands program.

At this level, as @Thomas Andrew said, it's certainly possible that some error has been missed. But in my opinion it would be merely an error, not a gap, because of the "coherence" of Wiles' proof. TNQD, September 20

Updated 10 Nov 2018

A proposed possible analytic logarithmic spiral solution to the Riemann hypothesis (abstract description)(Revised at 10 Nov 2018).

Preprint A proposed possible analytic logarithmic spiral solution to ...

note: I am looking for a collaboration and co authorship for a journal publication of the above proposal. You can reach me through Research Gate or directly via e-mail [email protected]

Preprint A proposed possible analytic logarithmic spiral solution to ...

For a positive logarithmic spiral located inside the positive area of the RH critical strip, at its singularity the spiraling curl tends to infinity proportional to the spiral parameter being b-->∞ with the complementary pitch φ -->0 and the spiral becomes a half-line or vector http://mathworld.wolfram.com/Ray.html with B the singularity origin in the critical line and A, its end point of the vector extending to +∞ .

The same process can be repeated for the negatve strip.

Therefore all complex numbers x±iy in the logarithmic spiraling critical strip are translated to points on the RH critical line thus all complex numbers of the the critical strip where ζ(χ±iy)=0 are points of the critical line.

copyright©Emmanouil Markoulaks Technological Educational Institute of Crete 2018

Few mathematicians are convinced by this. About a year ago Atiyah also circulated an alleged short proof of the Feit-Thompson theorem in group theory which came nowhere near to convincing anyone. So it looks as if it's becoming a habit. A sad ending to a great career.

Preprint A proposed possible analytic logarithmic spiral solution to ...

For a positive logarithmic spiral located inside the positive area of the RH critical strip, at its singularity the spiraling curl tends to infinity proportional to the spiral parameter being b-->∞ with the complementary pitch φ -->0 and the spiral becomes a half-line or vector http://mathworld.wolfram.com/Ray.html with B the singularity origin in the critical line and A, its end point of the vector extending to +∞ .

The same process can be repeated for the negatve strip.

Therefore all complex numbers x±iy in the logarithmic spiraling critical strip are translated to points on the RH critical line thus all complex numbers of the the critical strip where ζ(χ±iy)=0 are points of the critical line.

copyright©Emmanouil Markoulaks Technological Educational Institute of Crete 2018

Few mathematicians are convinced by this. About a year ago Atiyah also circulated an alleged short proof of the Feit-Thompson theorem in group theory which came nowhere near to convincing anyone. So it looks as if it's becoming a habit. A sad ending to a great career.

Preprint A proposed possible analytic logarithmic spiral solution to ...

For a positive logarithmic spiral located inside the positive area of the RH critical strip, at its singularity the spiraling curl tends to infinity proportional to the spiral parameter being b-->∞ with the complementary pitch φ -->0 and the spiral becomes a half-line or vector http://mathworld.wolfram.com/Ray.html with B the singularity origin in the critical line and A, its end point of the vector extending to +∞ .

The same process can be repeated for the negatve strip.

Therefore all complex numbers x±iy in the logarithmic spiraling critical strip are translated to points on the RH critical line thus all complex numbers of the the critical strip where ζ(χ±iy)=0 are points of the critical line.

copyright©Emmanouil Markoulaks Technological Educational Institute of Crete 2018

:)

Great approach!! However, I think this only proves that there are infinitely many roots on the half-line... You need to prove that there are none outside this line also. For example not all points on the half-line are roots. If you can prove that the singularities of the spirals are the roots then you have proved that all roots line on the half-line, except for the infinite root which is arbitrarily close to the half-line but not necessarily on the half-line since the singularity cannot be an end point also. For example I proved that s/(s-1) is an exponential function, which proves that s/(1-s-)=(1/2+1/2*i*tan(theta))/(1/2-1/2*i*tan(theta)) the half-line....But not accepted..

Fascinating!

Michael since for Re(x)=1 gives ζ(χ)= ∞ this is the boundary of critical strip and isolates the critical vortex strip from the rest of the numbers. Therefore I can whirl the whole critical strip around the symmetry axis critical line 1/2. All numbers are whirling into the singularity critical line 1/2 and becoming its points including all the roots of the zeta function.

I don't just spiral some points but the whole volume of the critical strip excluding the boundaries.

RH hypothesis can not be ever solved in 2D it must be transformed in 3D to do so.

I follow answers

best regards

Look at the fixed point theorem. No matter what your transformation may be there is always a fixed point. That point cannot be a root.

Dear Michael,

The only fixed points here are all the points in the critical line 1/2.

Essentially It is dilation transformation where the spiral it is reduced into a singularity, fixed point at the critical line 1/2.

I believe my analysis although possible many things are not yet covered has the potential to evolve into the solution.

In a spiral all the points fall into the singularity which can no other be located than at the 1/2 critical line.

I don't care which of these points at the critical line are actual roots of the zeta function since this was not a requirement of the RH hypothesis.

Kind Regards,

Emmanouil Markoulakis,

TEI of Crete

I think the fixed points of the spiral are the sources of the spiral and infinity.

I want to express my feelings based on my experience. After the age of eighty, we begin to be in a hurry to publish his papers for fear of a quick end of life. And we forget "In prudence safety and in haste regret". In any case, Sir Michael Atiyah has a great career.

I am still not sure whether this proof is right or wrong.

I myself would be surprised if it would be! hehe! just kidding...

Dear Dr. Tapas Chatterjee ,

As you probably are aware of, sometimes mathematical problems can have physical solution.

Riemann hypothesis is essentially describing logarithmic spirals therefore it seems natural to me the solution to involve vortices. What more natural to wrap everything with a vortex into a singularity. If symmetry holds, then the singularities can only be at the critical axis 1/2. Any function or physical process wrapped up at a singularity becomes infinite small thus essentially zero I submit.

Kind Regards,

Emmanouil

The axioms employed cannot be used to prove all cases. Kurt Godel. The proof can be valid for some roots but not for all. Anyone can prove that 1=2 but the fact that zero is employed nullifies his proof as substantially covering the space of all roots.

All the nontrivial zeros of zeta(s) function all have real part sigma=R[s]=1/2, the "critical line."

So what is the problem with the singularity being infinite small thus zero? Therefore possible proving the RH hypothesis?

If you refer to that all "vortices terminate to a singularity" as an axiom subject to Gödel's incompleteness theorem, I hardly can see this as an axiom used in this proof.

Anyway, I will not resolve to defend this prove. I will be more than happy if it can be thoroughly mathematically disproved! I will learn therefore something new and correct upon it the proposed solution.

Sadly, Sir Michael died yesterday. He was a truly great mathematician.

My condolances. Science community has lost one of its bests.

Emmanuel I was referring to Sor Michael.

okay.

So we'll never know the long and short of Atiyah's alleged demonstration of the RH. But, in spite of the skepticism of most experts, there is no doubt that it will soon join up, in the legend of math, Fermat's cryptic note in the margin of his copy of the Arithmetica of Diophantus : " I have discovered a truly marvelous proof..."

Yes there is no actual official paper published about his RH solution he can take credit of. He said in his presentation that he could not publish even at Arxiv! They did not accept it!!...

Since there is no formal complete manuscript of his RH solution I guess he took his secret on his grave!

The iterated exponential converges in the Shell-Thron region. Dropping onto the Reals, this gives convergence for (1/e)^e

Interesting latest development on the RH hypothesis:

https://www.pnas.org/content/early/2019/05/20/1902572116

How spiral lines can appear being straight lines:

https://tinyurl.com/y4egnykn