Nigerian Mathematician Enoch Opeyemi holds the most plausible solution to the Riemann Hypothesis.
Dr Nina Ringo, a Russian mathematician who is a specialist in the field of mathematical control theory and an organizer of the International Conference on Mathematics and Computer Science, also believes that the German Problem has found a solid resolution in Nigeria. Dr Opeyemi presented convincing results to a community of renowned mathematicians and many scientists in Africa are enamored by the elegance of his methods.
In a phone conversation with Dr Ringo, she informed Grandmother that the official release of the results obtained by Dr Opeyemi will be forthcoming.
However, many western scientists are skeptical about the proof, as an African mathematician – working from the humble space of an African College – has clearly beaten them to the solution.
The Riemann Hypothesis is one of the long standing problems in mathematics which has a one million dollar prize tag (offered by the Clay Mathematics Institute in Massachusetts) attached to a proof. It was first formulated by the very influential German mathematician Bernhard Riemann in 1859.
A senior lecturer at the Federal University in Oye-Ekiti, Nigeria, Enoch Opeyemi announced a proof at a talk during the International Conference on Mathematics and Computer Science (ICMS 2015) held in Vienna from November 10-11.
This announcement has generated a lot of buzz in the media and some skeptical reactions bordering on credulous hostility in some sections of the academic community – particularly western academia. What exactly is the Riemann hypothesis? Why it is important and what are the profound implications if Dr Opeyemi’s proof outlasts the horrid criticisms?
We all remember from grade school mathematics what exponents are. For example, 3 raised to the exponent 2 (32) is the same as 3 X 3 = 9. In the field of complex analysis that deals with the mathematics of complex numbers, we can define and make sense of a number raised to a complex exponent. This means that the exponent is a complex number. Complex numbers are special numbers that are written in the form a + ib where i has the special property that i raised to the exponent 2 equals -1, where a and b are just any real numbers.
Now we can introduce another concept from grade school mathematics, the concept of the reciprocal of a number. For example, the reciprocal of the number 2 is 1/2 (the fraction, a half), the reciprocal of 32 is 1/9. Now take any natural number and raise it to a complex exponent which we will denote by z (note that z = a + ib), the natural number n raised to a complex exponent is written as nz. Now take the reciprocal of this natural number n raised to the complex exponent z. Now we know that the natural numbers are infinite, if I start counting from 0, 1, 2, 3… I can go on and on all the way to infinity.
We add up all these reciprocals of all natural numbers raised to the complex exponent z. It is the infinite sum of all these reciprocals that we call the Riemann Zeta Function. In mathematical terms, we say that it is an absolutely convergent series and analytic on a certain region of the plane. Basically we take this Riemann Zeta function and then extend it by a process called analytic continuation such that it is defined on the entire complex plane except at z = 1 where it blows up. What this means can be understood with this analogy below:
Think of the complex plane as a flat map with an origin and your North, South, East and West as your cardinal directions.
With this image of the complex plane in mind, let’s examine another concept from grade school mathematics. That is, the solutions of equations. You might recall that the simple equation x-2 = 0 has a solution x=2. Another simple equation is a quadratic equation of the form (x-2)(x-3) = 0 which has solutions x =2 and x= 3.
Mathematicians like to generalize things since it makes their study much more interesting. We can introduce the notation f(x) = x- 2 and also similarly f(x) = (x-2)(x-3). I can then say that for the equation f(x) = 0 [where f(x) = x-2], x = 2 is a solution. We call the solution x = 2 a zero of the function f(x) = x-2. Similarly for the equation f(x) = 0 [where f(x) = (x-2)(x-3)], x = 2 and x= 3 are solutions. Again we call the solutions x = 2, and x = 3 zeros of the function f(x) = (x-2)(x-3).
We can now appreciate the Riemann zeta function by this convention. This Riemann Zeta function has points in the complex plane where its value is zero or in other words where it vanishes. We can think of it this way, the value that a function takes at a certain point on the complex plane (or map) is the echo of the function at that point. If the function at a certain point vanishes, we can say that it has no echo (the echo is equal to zero). If the function blows up a point, we say that it has an infinitely loud echo at that point.
The German mathematician Bernhard Riemann claimed in 1859 that this special function, the Riemann zeta function vanishes at non-trivial points on a certain line in the complex plane. We will use the analogy of echoes of the function to explain what this means.
Imagine that you are on your flat map at the origin. You now move 1/2 units due east along the East-West axis. After you move the 1/2 units due east, you then move along a vertical line up due north or down due south along the North-South axis. Riemann claims that as you move along this vertical line, there are points on this line where the zeta function has no echo at all (that is, its value is zero). He in fact claims that all the non-trivial points on the map where the zeta function has no echo lie on this vertical line. We note for mathematical completeness sake that, the zeta function has an infinitely loud echo 1 unit due east from the origin along the East-West axis.
So why is it important that the Riemann zeta function vanishes at non-trivial points on this special vertical line? This was because, Riemann made an important discovery about the connection between the non-trivial zeros of the Riemann zeta function and the distribution of prime numbers.
Prime numbers are numbers which are divisible by 1 and itself. Prime numbers are the building block of natural numbers. We can decompose any natural number as a product or primes or primes raised to a positive integer exponent. Primes are very rare beasts. As you move through the positive integers, prime number become harder to find. They become rare as platinum or diamond.
Riemann was trying to find a formula for the number of primes less than any given number. So basically, he was trying to answer the question, given any real number x, how many primes are there that are less than x? In his formula that he found to answer this interesting question, he realized that the zeroes of the Riemann zeta function were occurring in his explicit formula for the number of primes less than any given number x.
Therefore, using our analogy to the echoes of the Riemann zeta function on the map, the non-trivial points on the vertical line running North-South and a half unit due east from the origin where the Riemann zeta function has no echo are deeply connected to determining how many primes that are less than a given number x.
Another way of saying this is that, the distribution of prime numbers on the number line is controlled by the zeroes of the Riemann zeta function. This was certainly a deep connection that Riemann made. This is all good and aesthetically beautiful from a purely mathematical point of view. But what are the applications?
One important application of the distribution of primes on the number line is in Cryptography, the mathematics of encryption which makes sure that your communications stay safe and your credit card or debit card transactions are safe and sound. The security of many encryption algorithms is based on the fact that it is very easy to multiply two large prime numbers, but it is computationally intensive to do the reverse that is, given a large number, find the two large prime factors whose product is that number. This is because as we said, primes become rare as diamonds the further you move along the number line, and the distribution of primes on the number line is controlled by the zeroes of the Riemann zeta function.
There have been several attempts to prove the Riemann Hypothesis. Many methods of attack have been proposed to tackle this difficult problem. Notable mathematicians like David Hilbert and George Polya earlier on in the 20th century suggested the use of operator methods to prove the Riemann Hypothesis. The idea was to find a matrix representation (self-adjoint operator to be more exact) of the Riemann zeta function and then from spectral analysis of the spectrum of that operator to find the location of the zeroes of the Riemann zeta function.
At the ICMS 2015, Dr Enoch Opeyemi presented preliminary results on the spectrum of a matrix representation of the Riemann zeta function and derived some important conclusions which appear to be the most plausible solution to the Riemann problem ever conceived.
As we wait for the official release of his work, we are pleased with Dr Opeyemi’s aptitude. If his solutions outlast the myriad criticisms, it will confirm one of the major triumphs of mathematics in the last 200 years.
For this to have an African imprimatur will be, all the more, a crowning achievement to an Africa awakening from centuries of intellectual slumber. Africa might yet regain her stature in the rarefied world of high intellect and abstract thought, the empire of the mind.
This is an elegant piece by Jehuti Nefekare about an elegant mathematician in Enoch Opeyemi. Need I say more?
I enjoyed reading this essay on the Riemann Hypothesis and Enoch’s proof. Great contribution by Jehuti Nefekare!
Its false unfortunately
Have you seen the proof Ayelam Valentine Agaliba?
Apparently, he is a well known academic plagiarist/frauster. Anyway Check the website of the awarding body. Give me a moment. I will post a link below: http://www.claymath.org/millennium…/riemann-hypothesis.
I see, Ayelam Valentine Agaliba! So you actually haven’t seen the proof by Prof. Enoch Opeyemi. Thanks!
Ayelam, what is the proof that he is a fraudster. Grisha Perelman posted his papers on the proof of the poincare conjecture online. He did not submit it to the clay Institute. My advisor in grad school made me read his two fundamental papers on the Ricci flow that established the poincare conjecture. The preprints were available in the community. It was only after a few years that the clay Institute got involved after the proof stood the test. So the clay can’t say it is false without seeing the preprints.
Malcolm describes the difference between the “House Negro” and the “Field Negro” in a Michigan State University, East Lansing, speech on 23 January 1963:
So you have two types of Negro. The old type and the new type. Most of you know the old type. When you read about him in history during slavery he was called “Uncle Tom.” He was the house Negro. And during slavery you had two Negroes. You had the house Negro and the field Negro.
The house Negro usually lived close to his master. He dressed like his master. He wore his master’s second-hand clothes. He ate food that his master left on the table. And he lived in his master’s house–probably in the basement or the attic–but he still lived in the master’s house.
So whenever that house Negro identified himself, he always identified himself in the same sense that his master identified himself. When his master said, “We have good food,” the house Negro would say, “Yes, we have plenty of good food.” “We” have plenty of good food. When the master said that “we have a fine home here,” the house Negro said, “Yes, we have a fine home here.” When the master would be sick, the house Negro identified himself so much with his master he’d say, “What’s the matter boss, we sick?” His master’s pain was his pain. And it hurt him more for his master to be sick than for him to be sick himself. When the house started burning down, that type of Negro would fight harder to put the master’s house out than the master himself would.
But then you had another Negro out in the field. The house Negro was in the minority. The masses–the field Negroes were the masses. They were in the majority. When the master got sick, they prayed that he’d die. [Laughter] If his house caught on fire, they’d pray for a wind to come along and fan the breeze.
If someone came to the house Negro and said, “Let’s go, let’s separate,” naturally that Uncle Tom would say, “Go where? What could I do without boss? Where would I live? How would I dress? Who would look out for me?” That’s the house Negro. But if you went to the field Negro and said, “Let’s go, let’s separate,” he wouldn’t even ask you where or how. He’d say, “Yes, let’s go.” And that one ended right there.
So now you have a twentieth-century-type of house Negro. A twentieth-century Uncle Tom. He’s just as much an Uncle Tom today as Uncle Tom was 100 and 200 years ago. Only he’s a modern Uncle Tom. That Uncle Tom wore a handkerchief around his head. This Uncle Tom wears a top hat. He’s sharp. He dresses just like you do. He speaks the same phraseology, the same language. He tries to speak it better than you do. He speaks with the same accents, same diction. And when you say, “your army,” he says, “our army.” He hasn’t got anybody to defend him, but anytime you say “we” he says “we.” “Our president,” “our government,” “our Senate,” “our congressmen,” “our this and our that.” And he hasn’t even got a seat in that “our” even at the end of the line. So this is the twentieth-century Negro. Whenever you say “you,” the personal pronoun in the singular or in the plural, he uses it right along with you. When you say you’re in trouble, he says, “Yes, we’re in trouble.”
But there’s another kind of Black man on the scene – like Ayelam Valentine Agaliba. If you say you’re in trouble, he says, “Yes, you’re in trouble.” [Laughter] He doesn’t identify himself with your plight whatsoever!
Are you not being a little hard on young Ayelam Valentine Agaliba, Solomon Azumah-Gomez? Granted, he does have a tendency to intellectualize the most inappropriate of issues, reducing the arguments to the most simplistic of formulaic of strands, no doubt to impress or intimidate the uninitiated. It is further accepted that the young man frequently makes pronouncements with levels of certainty that truly accomplished philosophers would naturally hedge around with caveats; but all this is probably a function of youthful exuberance, rather than a wilful desire to use the analytical tools he has clearly acquired, to further some extraneous interest. That would be too depressing in one so young and full of promise. Perhaps, in another 10 or so years, when the real world has sufficiently impinged on his consciousness, the tools will merge with the skills to enable him make a more meaningful contribution to these conversations. Let’s give him the benefit of the doubt and not call him a ‘house Negro’ just yet!
You amuse me …;
Rather than correct our brother Solomon for his purile rant here you are doing your lawyer thing. I am not sure who is worst, you or him?
I fully retract my comment Fifi. Well said. I am sorry to have called you a Houe Negro Ayelam Valentine Agaliba. Please forgive me! I shall wait and see in what direction this evolutionary process carries you. Another 10 or so years…
I love it when civility triumphs over intemperate language. Perhaps we can make some real progress with the subject in hand.
Same with Andrew Wiles proof of the fermat last theorem, he announced a proof at a conference, same thing that Opeyemi did. Later Wiles proof was found to have a flaw, so he went back to work on it before it was finally accepted. It can take years for this kind of stuff. So for the clay to dismiss it without having mathematicians pour over the proof in his preprints is stupid.
Good point Jonathan Nukpezah. Let us hope that his results if indeed they constitute a proof stand the test.
And believe me, once the preprints come out, people will be going over it with a tooth pick. The methods he described in his abstract is real math. He is studying the point spectra of a self adjoint operator which is a representation of the Riemann zeta function. Hilbert suggested such an approach a long time ago.
In African American: HATERS GONN’ HATE!
Before you condemn the solution as a hoax you need to read this article!
I loved this piece by Jehuti Nefekare. It was pure and simple. We all wait for Dr Opeyemi’s final release of this proof. This will be a stupendous achievement especially since the few Russian mathematicians who have seen the solution think it’s elegant. But we wait.
I am really excited about what this would mean to the academic community in Africa and how ti would inspire more of the younger generation to venture into mathematics and science.
Yes Sir! Even more, I think, it will bring Mathematics back to where it originated as an art form. For example, the Dogons do this in their sleep, it’s like a drum beat that never stops, a rhythm that never dies. Hopefully, even though I think that Grandmother Africa’s source is legit, Dr Opeyemi has solved this problem. It will shame MIT, Harvard and Oxford all put together. This means a lot. It will rekindle something never before beheld in the eyes of today’s Africans. People are yet to see what this continent can really do. I sure look forward to his final presentation of the proof.
This is great news Atiga. The only way to deal with detractors is by showing them what we are capable of not endless complaining. Civilisation began in Africa and it is no secret that our ancestors were mathematical and astronomy geniuses to name a few.
We may have had a lot of setbacks but our time to shine forth the glory that is in us is now and I believe that in the coming years thousands of individuals like this are going to arise from all fields on this our continent.
Such is the Case!! Bravo fellow, I believe the hour and the time had come for Africa to begin redefining our ancient scholastic prowes on the continent again.
This suppose not to be strange because it is inborn but have been robbed of our history to acknowledged who we are as an intellectual spiritual being.
Congrat Dr.Enoch Opeyemi….. the sun have chosen to rise and set in Africa period.