Michael Francis Atiyah, a professor at Oxford, Cambridge and Edinburgh Universities and winner of almost a dozen prestigious awards in mathematics, presented a proof of the Riemann Hypothesis, one of the seven Millennium Problems, which describes how the prime numbers are located on the number line.
Atiyah's proof is short, taking up five pages, together with the introduction and bibliography. The scientist claims that he found a solution to the hypothesis by analyzing the problems associated with the fine structure constant, and used the Todd function as a tool. If the scientific community considers the proof correct, then the Briton will receive $ 1 million for it from the Clay Mathematics Institute (Clay Mathematics Institute, Cambridge, Massachusetts).
Other scientists are also vying for the prize. In 2015, he announced the solution of the Riemann hypothesis Professor of Mathematics Opeyemi Enoch from Nigeria, and in 2016 presented his proof of the hypothesis Russian mathematician Igor Turkanov. According to representatives of the Institute of Mathematics, in order for the achievement to be recorded, it must be published in an authoritative international journal, followed by confirmation of the proof by the scientific community.
What is the essence of the hypothesis?
The hypothesis was formulated back in 1859 by the German mathematician Bernhard Riemann. He defined a formula, the so-called zeta function, for the number of primes up to a given limit. The scientist found that there is no pattern that would describe how often prime numbers appear in the number series, while he found that the number of prime numbers that do not exceed x, is expressed in terms of the distribution of the so-called "non-trivial zeros" of the zeta function.
Riemann was confident in the correctness of the derived formula, but he could not establish on what simple statement this distribution completely depends. As a result, he put forward the hypothesis that all non-trivial zeros of the zeta function have a real part equal to ½ and lie on the vertical line Re=0.5 of the complex plane.
The proof or refutation of the Riemann hypothesis is very important for the theory of the distribution of prime numbers, says PhD student of the Faculty of Mathematics of the Higher School of Economics Alexander Kalmynin. “The Riemann Hypothesis is a statement that is equivalent to some formula for the number of primes not exceeding a given number x. A hypothesis, for example, allows you to quickly and with great accuracy calculate the number of prime numbers that do not exceed, for example, 10 billion. This is not the only value of the hypothesis, because it also has a number of rather far-reaching generalizations, which are known as the generalized Riemann hypothesis , the extended Riemann hypothesis, and the grand Riemann hypothesis. They are even more important for different branches of mathematics, but first of all, the importance of a hypothesis is determined by the theory of prime numbers,” says Kalmynin.
According to the expert, with the help of a hypothesis, it is possible to solve a number of classical problems of number theory: Gauss problems on quadratic fields (the problem of the tenth discriminant), Euler's problems on convenient numbers, Vinogradov's conjecture on quadratic non-residues, etc. In modern mathematics, this hypothesis is used to prove statements about prime numbers. “We immediately assume that some strong hypothesis like the Riemann hypothesis is true, and see what happens. When we succeed, we ask ourselves: can we prove it without assuming a hypothesis? And, although such a statement is still beyond what we can achieve, it works like a beacon. Due to the fact that there is such a hypothesis, we can see where we are going,” says Kalmynin.
The proof of the hypothesis can also affect the improvement of information technology, since the processes of encryption and coding today depend on the effectiveness of different algorithms. “If we take two simple large numbers of forty digits and multiply, then we will get a large eighty-digit number. If we set the task to factorize this number, then this will be a very complex computational task, on the basis of which many information security issues are built. All of them consist in creating different algorithms that are tied to the complexities of this kind, ”says Kalmynin.
The 15-line solution was presented by the famous British scientist Sir Michael Francis Atiyah ( Michael Francis Atiyah), winner of prestigious mathematical awards. He mainly works in the field of mathematical physics. Science reports that Atiyah spoke about his discovery at a conference Heidelberg Laureate Forum at Heidelberg University on Monday.
The Riemann hypothesis was formulated, as you might guess, by Bernhard Riemann in 1859. The mathematician introduced the concept of the zeta function - a function for a complex variable - and used it to describe the distribution of prime numbers. The original problem with primes was that they are simply distributed over a series of natural numbers without any apparent pattern. Riemann proposed his distribution function for prime numbers not exceeding x, but he could not explain why the dependence arises. Scientists have been struggling to solve this problem for almost 150 years.
The Riemann hypothesis is included in the list of "" (Millennium Prize Problems), for the solution of each of which a million dollar reward is due. Of these problems, only one has been solved - the Poincare conjecture. Its solution was proposed by a Russian mathematician back in 2002 in a series of his papers. In 2010, the scientist was awarded the prize, but he refused it.
Michael Atiyah claims to have explained Riemann's pattern. In his proof, the mathematician relies on the fundamental physical constant - the fine structure constant, which describes the strength and nature of electromagnetic interactions between charged particles. Describing this constant using the relatively obscure Todd function, Atiyah found a solution to the Riemann hypothesis by contradiction.
The scientific community is in no hurry to accept the proposed proof. For example, an economist from the Norwegian University of Science and Technology Jørgen Visdal ( Jørgen Veisdal), who had previously studied the Riemann Hypothesis, stated that Atiyah's solution was "too vague and uncertain". The scientist needs to study the written evidence more carefully in order to come to conclusions. Atiyah's colleagues contacted Science, also noted that they do not consider the presented solution to be successful, since it is based on shaky associations. UC Riverside mathematical physicist John Baez ( John Baez) and even stated that Atiyah's proof "simply imposes one impressive claim on another without any arguments in favor of it or real justifications."
Michael Atiyah himself believes that his work lays the groundwork for proving not only the Riemann Hypothesis, but also other unresolved problems in mathematics. Of criticism, he says, "People will complain and grumble, but that's because they don't agree with the idea that the old man could come up with a whole new method."
Interestingly, in the past, the scientist has already made similar high-profile statements and faced criticism. In 2017, Atiyah told the London edition The Times that he reduced the 255-page Feit-Thompson or Odd Order Theorem, proved in 1963, to 12 pages. The mathematician sent his proof to 15 experts, but they never gave positive marks to the work, and as a result, it was not published in any scientific journal. A year earlier, Atiyah had announced the solution of a well-known problem in differential geometry. The scientist published a preprint of the article with this solution on ArXiv.org. Soon, colleagues pointed out a number of inaccuracies in the work, and the full-text version of the article was never published.
These errors now largely support the scientific community's skepticism about proving the Riemann Hypothesis. Atiye has to wait for the assessment of the Clay Institute, which gives out awards for solving the “millennium problems”. For now, you can read the proof of the mathematician at the link to Google Drive, which he himself posted in the public domain.
Hello, habralyudi!
Today I would like to touch upon such a topic as the “millennium tasks”, which have been worrying the best minds of our planet for decades, and some even hundreds of years.
After proving the conjecture (now the theorem) of Poincaré by Grigory Perelman, the main question that interested many was: “ And what did he actually prove, explain on your fingers?» Taking the opportunity, I will try to explain on my fingers the other tasks of the millennium, or at least approach them from another side closer to reality.
Equality of classes P and NP
We all remember quadratic equations from school, which are solved through the discriminant. The solution to this problem is class P (P olynomial time)- for it, there is a fast (hereinafter, the word "fast" is meant as executing in polynomial time) solution algorithm, which is memorized.There are also NP-tasks ( N on-deterministic P olynomial time), the found solution of which can be quickly checked using a certain algorithm. For example, check by brute-force computer. If we return to the solution of the quadratic equation, we will see that in this example the existing solution algorithm is checked as easily and quickly as it is solved. From this, a logical conclusion suggests itself that this task belongs to both one class and the second.
There are many such tasks, but the main question is whether all or not all tasks that can be easily and quickly checked can also be easily and quickly solved? Now, for some problems, no fast solution algorithm has been found, and it is not known whether such a solution exists at all.
On the Internet, I also met such an interesting and transparent wording:
Let's say that you, being in a large company, want to make sure that your friend is also there. If you are told that he is sitting in the corner, then a fraction of a second will be enough to, with a glance, make sure that the information is true. In the absence of this information, you will be forced to go around the entire room, looking at the guests.
In this case, the question is still the same, is there such an algorithm of actions, thanks to which, even without information about where a person is, find him as quickly as if knowing where he is.
This problem is of great importance for various fields of knowledge, but it has not been solved for more than 40 years.
Hodge hypothesis
In reality, there are many simple and much more complex geometric objects. Obviously, the more complex the object, the more time-consuming it becomes to study. Now scientists have invented and are using with might and main an approach, the main idea of which is to use simple "bricks" with already known properties that stick together and form its likeness, yes, a designer familiar to everyone since childhood. Knowing the properties of the "bricks", it becomes possible to approach the properties of the object itself.Hodge's hypothesis in this case is connected with some properties of both "bricks" and objects.
Riemann hypothesis
Since school, we all know prime numbers that are divisible only by itself and by one. (2,3,5,7,11...) . Since ancient times, people have been trying to find a pattern in their placement, but luck has not smiled at anyone so far. As a result, scientists have applied their efforts to the prime number distribution function, which shows the number of primes less than or equal to a certain number. For example, for 4 - 2 prime numbers, for 10 - already 4 numbers. Riemann hypothesis just sets the properties of this distribution function.Many statements about the computational complexity of some integer algorithms are proven under the assumption that this conjecture is true.
Yang-Mills theory
The equations of quantum physics describe the world of elementary particles. Physicists Yang and Mills, having discovered the connection between geometry and elementary particle physics, wrote their own equations, combining the theories of electromagnetic, weak and strong interactions. At one time, the Yang-Mills theory was considered only as a mathematical refinement, not related to reality. However, later the theory began to receive experimental confirmation, but in general it still remains unresolved.On the basis of the Yang-Mills theory, the standard model of elementary particle physics was built within which the sensational Higgs boson was predicted and recently discovered.
Existence and smoothness of solutions of the Navier-Stokes equations
Fluid flow, air currents, turbulence. These and many other phenomena are described by equations known as Navier-Stokes equations. For some special cases, solutions have already been found in which, as a rule, parts of the equations are discarded as not affecting the final result, but in general the solutions of these equations are unknown, and it is not even known how to solve them.Birch-Swinnerton-Dyer hypothesis
For the equation x 2 + y 2 \u003d z 2, Euclid once gave a complete description of the solutions, but for more complex equations, finding solutions becomes extremely difficult, it is enough to recall the history of the proof of Fermat's famous theorem to be convinced of this.This hypothesis is connected with the description of algebraic equations of the 3rd degree - the so-called elliptic curves and is in fact the only relatively simple general way to calculate rank, one of the most important properties of elliptic curves.
In proof Fermat's theorems elliptic curves have taken one of the most important places. And in cryptography, they form a whole section of the name itself, and some Russian digital signature standards are based on them.
Poincare conjecture
I think if not all, then most of you have definitely heard about it. Most often found, including in the central media, such a transcript as “ a rubber band stretched over a sphere can be smoothly pulled to a point, but a rubber band stretched over a donut cannot". In fact, this formulation is valid for the Thurston conjecture, which generalizes the Poincaré conjecture, and which Perelman actually proved.A special case of the Poincare conjecture tells us that any three-dimensional manifold without boundary (the universe, for example) is like a three-dimensional sphere. And the general case translates this statement to objects of any dimension. It is worth noting that a donut, just like the universe is like a sphere, is like an ordinary coffee mug.
Conclusion
At present, mathematics is associated with scientists who have a strange appearance and talk about equally strange things. Many talk about her isolation from the real world. Many people of both younger and quite conscious age say that mathematics is an unnecessary science, that after school / institute, it was not useful anywhere in life.But in fact, this is not so - mathematics was created as a mechanism with which to describe our world, and in particular, many observable things. It is everywhere, in every home. As V.O. Klyuchevsky: “It’s not the flowers’ fault that the blind man doesn’t see them.”
Our world is far from being as simple as it seems, and mathematics, in accordance with this, is also becoming more complex, improving, providing more and more solid ground for a deeper understanding of the existing reality.
Russian mathematician found proof of the Riemann Hypothesis January 3rd, 2017
Bernhard Riemann
Remember, I told you about . So, among them was the Riemann hypothesis.
In 1859, the German mathematician Bernhard Riemann took Euler's old idea and developed it in a completely new way, defining the so-called zeta function. One result of this work was an exact formula for the number of primes up to a given limit. The formula was an infinite sum, but analysts are no strangers to that. And it was not a useless game of the mind: thanks to this formula, it was possible to obtain new genuine knowledge about the world of prime numbers. There was only one small problem. Although Riemann could prove that his formula was exact, the most important potential consequences of it depended entirely on one simple statement about the zeta function, and it was that simple statement that Riemann could never prove. A century and a half later, we still haven't managed to do it.
Today, this statement is called the Riemann hypothesis and is, in fact, the holy grail of pure mathematics, which seems to have "found" Russian mathematician.
This may mean that the world mathematical science is on the verge of an international event.
The proof or refutation of the Riemann hypothesis will have far-reaching consequences for number theory, especially in the field of the distribution of prime numbers. And this can affect the improvement of information technology.
The Riemann Hypothesis is one of the seven Millennium Problems, for which the Clay Mathematics Institute (Cambridge, Massachusetts) will pay a reward of one million US dollars for solving each of them.
Thus, the proof of the conjecture can enrich the Russian mathematician.
According to the unwritten laws of the international scientific world, Igor Turkanov's success will not be fully recognized until a few years later. However, his work has already been presented at the International Physics and Mathematics Conference under the auspices of the Institute of Applied Mathematics. Keldysh RAS in September 2016.
We also note that if the proof of the Riemann Hypothesis found by Igor Turkanov is recognized as correct, then the solution of two of the seven "millennium problems" will already be credited to the account of Russian mathematicians. One of these problems is the "Poincaré hypothesis" in 2002. At the same time, he refused the bonus of $1 million from the Clay Institute that was due to him.
In 2015, Professor of Mathematics Opeyemi Enoch from Nigeria claimed that he was able to solve the Riemann Hypothesis, but the Clay Institute of Mathematics considered the Riemann Hypothesis unproven until now. According to representatives of the institute, in order for the achievement to be recorded, it must be published in a reputable international journal, with subsequent confirmation of the proof by the scientific community.
sources
Mathematical science. Work on them had a tremendous impact on the development of this area of human knowledge. 100 years later, the Clay Mathematical Institute presented a list of 7 problems known as the Millennium Problems. Each of them was offered a prize of $1 million.
The only problem that appeared among both lists of puzzles that have been haunting scientists for more than one century was the Riemann hypothesis. She is still waiting for her decision.
Brief biographical note
Georg Friedrich Bernhard Riemann was born in 1826 in Hannover, in a large family of a poor pastor, and lived only 39 years. He managed to publish 10 works. However, already during his lifetime, Riemann was considered the successor of his teacher Johann Gauss. At the age of 25, the young scientist defended his dissertation "Fundamentals of the theory of functions of a complex variable." Later he formulated his hypothesis, which became famous.
prime numbers
Mathematics appeared when man learned to count. At the same time, the first ideas about numbers arose, which they later tried to classify. Some of them have been observed to have common properties. In particular, among the natural numbers, i.e., those that were used in counting (numbering) or designating the number of objects, a group was singled out that were divisible only by one and by themselves. They are called simple. An elegant proof of the theorem of infinity of the set of such numbers was given by Euclid in his Elements. On the this moment their search continues. In particular, the largest of the already known is the number 2 74 207 281 - 1.
Euler formula
Along with the concept of the infinity of the set of primes, Euclid also defined the second theorem on the only possible decomposition into prime factors. According to it, any positive integer is the product of only one set of prime numbers. In 1737, the great German mathematician Leonhard Euler expressed Euclid's first infinity theorem in the form of the formula below.
It is called the zeta function, where s is a constant and p takes on all prime values. Euclid's statement about the uniqueness of the expansion directly followed from it.
Riemann zeta function
Euler's formula, on closer inspection, is absolutely amazing, as it defines the relationship between primes and integers. After all, on its left side, infinitely many expressions that depend only on prime numbers are multiplied, and on the right side there is a sum associated with all positive integers.
Riemann went further than Euler. In order to find the key to the problem of the distribution of numbers, he proposed to define a formula for both real and complex variables. It was she who subsequently received the name of the Riemann zeta function. In 1859, the scientist published an article entitled "On the number of prime numbers that do not exceed a given value", where he summarized all his ideas.
Riemann suggested using the Euler series, which converges for any real s>1. If the same formula is used for complex s, then the series will converge for any value of this variable with a real part greater than 1. Riemann applied the analytic continuation procedure, extending the definition of zeta(s) to all complex numbers, but "thrown out" the unit. It was excluded because for s = 1 the zeta function increases to infinity.
practical meaning
A natural question arises: what is interesting and important about the zeta function, which is the key to Riemann's work on the null hypothesis? As you know, at the moment no simple pattern has been identified that would describe the distribution of prime numbers among natural numbers. Riemann was able to discover that the number pi(x) of primes that did not exceed x is expressed in terms of the distribution of non-trivial zeros of the zeta function. Moreover, the Riemann Hypothesis is a necessary condition for proving time estimates for the operation of some cryptographic algorithms.
Riemann hypothesis
One of the first formulations of this mathematical problem, which has not been proven to this day, sounds like this: non-trivial 0 zeta functions are complex numbers with a real part equal to ½. In other words, they are located on the line Re s = ½.
There is also a generalized Riemann hypothesis, which is the same statement, but for generalizations of zeta functions, which are usually called Dirichlet L-functions (see photo below).
In the formula χ(n) is some numerical character (modulo k).
The Riemannian assertion is considered the so-called null hypothesis, as it has been tested for consistency with existing sample data.
As Riemann argued
The remark of the German mathematician was initially formulated rather casually. The fact is that at that time the scientist was going to prove the theorem on the distribution of prime numbers, and in this context, this hypothesis did not have much meaning. However, its role in solving many other issues is enormous. That is why Riemann's assumption is currently recognized by many scientists as the most important of the unproven mathematical problems.
As already mentioned, to prove the distribution theorem, the full Riemann hypothesis is not needed, and it is enough to logically justify that the real part of any non-trivial zero of the zeta function is in the interval from 0 to 1. From this property it follows that the sum over all 0-th The zeta functions that appear in the exact formula above are a finite constant. For large values of x, it may be lost altogether. The only member of the formula that remains the same even for very large x is x itself. The remaining complex terms vanish asymptotically in comparison with it. So the weighted sum tends to x. This circumstance can be considered a confirmation of the truth of the theorem on the distribution of prime numbers. Thus, the zeros of the Riemann zeta function have a special role. It lies in the fact that the values cannot make a significant contribution to the decomposition formula.
Followers of Riemann
The tragic death from tuberculosis did not allow this scientist to bring his program to its logical end. However, Sh-Zh took over from him. de la Vallée Poussin and Jacques Hadamard. Independently of each other, they deduced a theorem on the distribution of prime numbers. Hadamard and Poussin succeeded in proving that all non-trivial 0 zeta functions are within the critical band.
Thanks to the work of these scientists, a new direction in mathematics appeared - the analytic theory of numbers. Later, several more primitive proofs of the theorem Riemann was working on were obtained by other researchers. In particular, Pal Erdős and Atle Selberg even discovered a very complex logical chain confirming it, which did not require the use of complex analysis. However, by this point, several important theorems had already been proved by means of Riemann's idea, including the approximation of many functions of number theory. In this regard, the new work of Erdős and Atle Selberg had practically no effect on anything.
One of the simplest and most beautiful proofs of the problem was found in 1980 by Donald Newman. It was based on the famous Cauchy theorem.
Does the Riemannian Hypothesis threaten the foundations of modern cryptography?
Data encryption arose along with the advent of hieroglyphs, more precisely, they themselves can be considered the first codes. At the moment, there is a whole area of digital cryptography, which is developing
Prime and "semi-prime" numbers, i.e. those that are only divisible by 2 other numbers in the same class, form the basis of the public key system known as RSA. It has the widest application. In particular, it is used when generating an electronic signature. Speaking in terms accessible to dummies, the Riemann hypothesis asserts the existence of a system in the distribution of prime numbers. Thus, the strength of cryptographic keys, on which the security of online transactions in the field of e-commerce depends, is significantly reduced.
Other unresolved mathematical problems
It is worth finishing the article by devoting a few words to other millennium tasks. These include:
- Equality of classes P and NP. The problem is formulated as follows: if a positive answer to a particular question is checked in polynomial time, is it true that the answer to this question itself can be found quickly?
- Hodge hypothesis. In simple words, it can be formulated as follows: for some types of projective algebraic varieties (spaces), Hodge cycles are combinations of objects that have a geometric interpretation, i.e., algebraic cycles.
- The Poincaré hypothesis. This is the only Millennium Challenge that has been proven so far. According to it, any 3-dimensional object that has the specific properties of a 3-dimensional sphere must be a sphere up to deformation.
- Statement of the quantum theory of Yang-Mills. It is required to prove that the quantum theory put forward by these scientists for the space R 4 exists and has a 0th mass defect for any simple compact gauge group G.
- Birch-Swinnerton-Dyer hypothesis. This is another issue related to cryptography. It concerns elliptic curves.
- The problem of the existence and smoothness of solutions of the Navier-Stokes equations.
Now you know the Riemann hypothesis. In simple terms, we have formulated some of the other Millennium Challenges. That they will be solved or it will be proved that they have no solution is a matter of time. Moreover, it is unlikely that this will have to wait too long, since mathematics is increasingly using the computing capabilities of computers. However, not everything is subject to technology, and first of all, intuition and creativity are required to solve scientific problems.
