If you check arixv you will find many proofs of RH and of non RH. I suppose the quoted paper appeared in an open access journal?

@Mohamed, Luiz, Andreas: thank you for your answers.

@Patrick: yes it is a paper from oa journal. In recent years i found numerous proofs on RH in arxiv and elsewhere, but apparently there is not yet a review on the status of such proofs. Thanks

Dear Luis Botelho,

Could you explain further your opinion on the proof of Fermat's Last Theorem ?

Dear all: i just posted a new question concerning PhotoMath, perhaps it can be used for solving Fermat Last theorem and other problems? 

https://www.researchgate.net/post/Have_you_tried_PhotoMath_Is_it_useful_for_you

Riemann's Hypothesis is not yet solved , except some ridiculous and  serious attempts to solve it !

Perhaps, it will exist some new proof " à la Kummer" with his regular primes related to the magical Bernoulli numbers...

Dear Luiz,

Thanks for clarifying your position. In return, let me explain my own views on « problem solving », « conjecture solving », and the development of mathematics (or any science) in general.

First let us go back to elementary school. I wonder if, when we were sweating on problems of train timetable or water leak, or other puzzles involving the rule of three, we were aware of learning how to solve linear equations – a technique which needed millenaries to be formalized, for instance in Al-Khwarizmi’s 825 treatise on « algebra » (one meaning of the word al-jabr is « to reset a fracture », here « to isolate an unknown quantity in order to solve it »). Later in college, we learned how to solve equations of the second (and perhaps also third and fourth) degree by « radicals », i.e. by calculating a discriminant and taking its square (or cubic) root. What’s my point ? It’s to stress that, even if we dispose of a « tool box » to « solve problems », first we must learn how to use these tools, second we are not assured that our box contains sufficiently many tools. Let’s go on to equations of degree five and beyond. « Solving the quintic » (by radicals) was a famous problem for centuries before it was shown to be impossible by Abel around 1820. But of course the real mathematical breakthrough on this subject was Galois theory, which shifted the focus from roots of polynomials to groups of permutations of these roots. Everybody knows to what extent the notion of groups has pervaded all the domains of science. That’s precisely my point : not only did Galois forge new tools, he actually built a big machine, so powerful that nowadays, in graduate courses on Galois theory, Abel’s theorem is reduced to a mere exercise :  Show that the alternate group A_5 is « simple » (in the sense of group theory). Aside the above algebraic examples, surely analysts and physicists can also rightly claim that the invention of « calculus » (differential and integral) drastically changed our comprehension of the physical world :  

Nature and Nature’s Laws lay hid in Night:

God said, “Let Newton be!” and all was light.

                                                       (Newton’s epitaph by Alexander Pope)

Granted (?) that mathematics are not a solidified, but an evolutive discipline (as all scientific disciplines must be), we can now get to the heart of our discussion : how do they evolve ? I see the researchers in mathematics, not as an army under orders, but as a bunch of individualistic explorers guided only by their taste and intuition, sometimes also by the visions of some « prophets » standing on top of high mountains. As real explorers in the real world, they can get blocked by swamps, rivers, canyons, but unlike real explorers, they can manage to go on by… making a conjecture. I must insist that not all mathematical problems, however difficult, deserve to be called conjectures. There is a degree in depth, pertinence, internal necessity for the development of mathematics. The so called « 4 colour problem » was solved in 1976 (with the aid of a computer), and the University of Illinois issued a special stamp on this occasion, but that was all it was. Because neither the question nor its solution did contribute in anything to the progress of our science. Without its integration into Galois theory, « solving the quintic » would have known the same fate. As for the Fermat equation… During his whole career, Gauss himself repeatedly and publicly said he was not interested by such an isolated problem. Strictly speaking, I think he was right. The Fermat equation has no trivial solution ? So what ? Just a riddle, in spite of its longevity and « romantic » aura. The fundamental importance of Wiles’ work of 500 pages (as you say) lies elsewhere. He proved the so called conjecture of Shimura-Taniyama-Weil on the « modularity of elliptic curves defined over the field Q of rational numbers ».

Dear Luiz, because you say you’re not an expert in number theory or algebraic geometry, let me try to explain very quickly (and hence very summarily !) what the STW conjecture is about. Everybody since elementary school knows the « 4 operations of arithmetic » : addition, substraction, multiplication, division. The late Martin Eichler is said to have proposed to add to them a 5th operation, the « modular forms » : these are special holomorphic functions on the complex upper half plane which possess the « fearful symmetries » of William Blake’s « Tyger » - so many symmetries one could doubt their existence. But they do exist ! Particular examples were known to Euler, Jacobi, Eisenstein etc. in relation with number theory, but their systematic study goes back only to the second half of the last century. The second « exotic bugs » coming in the STW conjecture are the « elliptic curves » : contrary to their name, these are not ellipses, but projective cubics (=algebraic curves of degree 3) which happen to possess a natural structure of commutative groups. Note that this group structure has been efficiently used in cryptography. When the elliptic curve E is defined over Q (this means roughly that it is defined by an equation with rational coefficients), one can attach to it a complex function L_E (s) which can be viewed as the analog of the zêta function (and we are back to our original subject !). The STW conjecture gives a precise link between between modular forms and elliptic curves over Q : roughly speaking, it says that the function L_E (s) is the inverse Mellin transform of a certain modular form (precise informations are given on that form). When you think of it, this is stupendous, because why would there be any link between a geometric object and an analyic object ? This is the same kind of mystery as the relation between the zêta function and the distribution of primes. Don’t you think that Wiles’ theorem deserves to be called the theorem of the 20th century ? It’s 20 years old now, and its offspring in numerous domains of number theory and algebraic geometry seems inexhaustible. Contrary to what you say, Wiles’ 500 pages are not devoted to the proof of a single result, they were necessary to develop a whole powerful machinery, just as in the case of Galois theory or calculus. These are now autonomous disciplines in graduate courses, and nobody would think to complain about the number of pages that students are obliged to swallow to master them.

One last word about Fermat’s equation. Contrary to what you say, the STW conjecture is not just « expected to be related » to FLT. As Abel’s theorem became just an exercise in Galois theory, FLT is now just a corollary of Wiles’ theorem. The « trick » is due to Hellegouarch and Frei : supposing that the Fermat equation has a non trivial solution (a, b, c), they construct an elliptic curve with rational coefficients involving a, b, c ; because of the symmetry of Fermat’s equation, this elliptic curve possesses extra properties which imply that it cannot be « modular » in the sense explained above. Thinking of it, the final proof of FLT must be the longest reductio at absurdum in the history of mathematics ! But I repeat my point : FLT is not important in itself, it’s STW which is important for the development of our science.

Hoping I have not been too boring or pedantic,     Thong  NQD

Dear Thong,

I think you are just too nice with people like Luiz. I compare them to people who think

that man has never landed on the Moon. Similarly you cannot check personally that the Apollo program was successful. However there is a mountain of evidence that the feat has been achieved. At least these conspiracy theorists have political excuse to be so, like anti-americanism and so on..

The excuses for not believing in STW are much weaker! Maybe it is an anti-establishment attitude?

Sincerely

Patrick

Dear Patrick,

How can you not believe in a mathematical proof ?!? It's there, and everybody can check it. The only cases where you "can't" :

- you are not qualified: then you must trust the experts who did the job. In the case of Wiles, there were 3 referees for his first version. They found a gap, he recognized it publicly, he filled the gap (with Taylor), he resubmitted his paper, etc. This is the normal procedure in the Republic of Science. Since the publication of the paper, there have been innumerable seminaries and conferences all over the world, and now books. Not to say anything of the papers coming after Wiles. In view of their number and value, any default (such as a contradiction) would have appeared

- you are not qualified, and you don't want to believe. Then it's no longer science, and no discussion is possible

Best,   Thong

@Luiz: using strong language does not make your point stronger. On the contrary!

Up to now, there is no real proof of Riemann' Hypothesis !

Serre and others mathematicians have already proved (ca .1980) that  Taniyama-Weil 's conjecture implies  FLT and Wiles- Taylor essentially proved that Taniyama-Weil is true.

Dear Zekiri,

You can delete the word "essentially" : Wiles (+Taylor ) showed the so called "semi-stable" case, and the rest was proved in full generality by Breuil-Conrad-Diamond6Taylor following Wiles' ideas .

Dear Luiz,

to answer your question peacefully, I do believe FLT has been proved twenty years ago. Of course I did not check every hairy detail myself but I do believe in the consensus of the experts (esp after twenty years and a Gauss medal). I also read Hellegouarch book at the time. Also attended a lecture by Serre.

Re your claim that 500 pages is too long, I have two observations. First its is very short compared to the classification of finite simple groups which is easily 100 times longer.Next, and most importantly, almost any result in graduate math level takes 500 pages to prove if you add up all the result you need to prove it! Think of Hahn-Banach theorem or Bolzano Weierstrass. You did not feel it when you were studying it because you learned it over the course of several years (say from age 15 to 25) but just have a look at the books in your library (and maybe in your attic!) and you will get my meaning.

Dear Luiz, dear Patrick,

1) To avoid any misunderstanding, I propose to remove the word "believe" from the mathematical (and more generally, scientific) language .

2) As for "book long" proofs, how about Perelman's proof of Poincaré's conjecture ? (see the 2 books written independently by 2 teams, american and chinese, who were specifically mandated for that task)

There are so many proofs on arxiv, vixra etc..Checking them all would be a full time job. It is fairly possible that one of these is correct but the author does not have the connections to be recognized by the "establishment"!

Luiz, are you able to discuss without insulting your interlocutor?

It's not proved yet, it seems that it will be considered as an Axiom!

Of course, the Riemann Hypothesis is true!  It's a SIN to think otherwise!  

Reference links: https://www.researchgate.net/publication/300068776_Proof_of_Riemann_Hypothesis;

https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true;  

or 

https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true2.

Working Paper Proof of Riemann Hypothesis

See appended file

See appended file

See appended file

Dear Prof. Thong,

I impressed by step two.

Best regards

Deleted research item The research item mentioned here has been deleted

Article Riemann Hypothesis Proof

Yes its solved.

Dear Keyvan Ghaffari

Your attempt is appreciated.

But it is not true to say it is solved unless you publish your article in a reputable journal.

Regards

Dear Victor Christianto Issam Kaddoura

Keyvan Ghaffari Thong Nguyen Quang Do

Short answer: not yet.

Many claim to have done so but after their proof is checked it always ends up being wrong. There is no officially accepted proof.

I am intersted in the (RH). How can I get updates related to the attempts of proving the (RH) @Roudy El Haddad , @Patrick Solé.

Yes its solved.

Preprint Short simple geometric Riemann Hypothesis Proof by convexity

What does the hypothesis say?

Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1/2 + i t, where t is a real number and i is the imaginary unit.

BUT, if we ask to prove whether the hypothesis is correct, we must ask for the values of the critical lines that confirm the Riemann hypothesis. Thus, the Riemann hypothesis is not true everywhere in the interval (0,1) with general ζ(q*z)=0.

There are solutions for non-trivial roots, but where ζ(s) is 1-1. Here we are considering a case, of the functional equations given to us by Riemann himself. But there is a generalization to this as well. This is another Hypothesis. Hypothesis for ζ(z)=0 is applied ie Re(z)=1/2 is correct.But for others conditions not apply as ζ(q*z)=0 we have other critical lines from (0,1),q>=1 or 0

Preprint Riemann Hypothesis Proof by convexity

Yes.

Dear Keyvan Ghaffari

Your efforts are appreciated. No mathematician dares to claim that he proved the Reimann Hypothesis unless they successfully published the results in a reputable Journal. Hundreds of proofs are available online, but all are not trusted before appearing in the right place.

I wish you good luck.

Interesting question for possible future fruitful discussions.

The Riemann Hypothesis is true. It is proven by me.

https://www.researchgate.net/publication/353014685_Riemann_Hypothesis_Proof_by_convexity