Compared with Fermat's conjecture, which took more than three and a half centuries to solve, and Goldbach's conjecture, which took more than two and a half centuries to survive, Riemann's conjecture is far from being recorded for only a century and a half, but its importance in mathematics is far greater than these two conjectures with higher public awareness. Riemann conjecture is the most important mathematical problem in mathematics today. At present, it is reported that Professor OpeyemiEnoch of Nigeria has successfully solved Riemann conjecture, but Clay Institute of Mathematics has neither confirmed nor denied that Dr Enoch has officially solved this problem.
An article on arxiv website points out that German mathematician C.L.Siegel proved Riemann conjecture in Riemann manuscript compiled by 1932. According to a conclusive formula in the manuscript, the author directly deduces that all zeros of the ζ(s) function in the rectangular region fall on the critical line.
The source of conjecture is Riemann conjecture, which was put forward by Riemann in 1859. The mathematician was born in 1826 in a small town called Bre Slentz, which now belongs to Germany and then to the kingdom of hanover. 1859, Riemann was elected as a member of the Communication Academy of Berlin. In return for this lofty honor, he submitted a paper entitled "On the number of prime numbers less than a given value" to the Berlin Academy of Sciences. This short eight-page paper is the birthplace of Riemann's conjecture.
Riemann conjecture researcher Riemann Riemann studied a problem that mathematicians have been interested in for a long time, that is, the distribution of prime numbers. A prime number is a number like 2,5, 19, 137 that cannot be divisible by other positive integers except 1 and itself. These numbers are very important in the study of number theory, because all positive integers greater than 1 can be expressed as their products. In a sense, their position in number theory is similar to that of atoms used to construct everything in the physical world. The definition of prime numbers is very simple, and they are taught in middle schools and even primary schools, but their distribution is unusual. Mathematicians have made great efforts, but they have not been fully understood so far.
An important achievement of Riemann's paper is that the mystery of prime number distribution is completely contained in a special function, especially a series of special points that make the value of that function zero have a decisive influence on the detailed law of prime number distribution. That function is now called Riemann zeta function, and that series of special points are called nontrivial zeros of Riemann zeta function.
Interestingly, although Riemann's article has made remarkable achievements, the text is extremely concise, even a little too concise, because it includes many places where "proof is omitted". Terrible is that the obvious proof should be omitted by "omitting proof", but Riemann's paper did not. Some of his "proof ellipsis" were completed by later mathematicians after decades of efforts, and some of them are still blank today. But in Riemann's paper, besides a lot of "proof is omitted", there is also a proposition that he clearly admits that he can't prove, that is, Riemann conjecture. Riemann conjecture: Since 1859 was born, 150 years have passed. During this period, it was like a towering mountain, which attracted countless mathematicians to climb, but no one could reach the top.
Of course, if we only compare it in time, this record of Riemann conjecture is far from being solved after three and a half centuries, and Goldbach conjecture has stood for more than two and a half centuries. But Riemann conjecture is far more important in mathematics than these two conjectures with higher public awareness. According to statistics, there are more than 1000 mathematical propositions based on Riemann conjecture (or its extended form) in today's mathematical literature. If Riemann conjecture is proved, those mathematical propositions can be promoted to theorems; On the other hand, if Riemann's conjecture is falsified, at least some of those mathematical propositions will be buried with him. It is extremely rare that a mathematical conjecture is closely related to so many mathematical propositions.
The equivalence theorem 190 1 helg von Koch points out that Riemann conjecture is equivalent to the prime number theorem with strong conditions.
Riemann observed that the frequency of prime numbers is closely related to the behavior of a well-constructed so-called Riemann zeta function zeta (). Riemann hypothesis asserts that all meaningful solutions of equation ζ(s)=0 are on a straight line. This has been verified in the original 1, 500,000,000 solutions.
Riemann zeta function zeta (s) is a series expression.
Analytic continuation on complex plane.
The reason why we want to generalize this expression analytically is that it only applies to the real part of S, the region of Re(S)>: 1 on the complex plane (otherwise the series will not converge). Riemann found the analytic extension of this expression (of course, Riemann did not use the modern terms of complex variable function theory such as "analytic extension"). Using path integration, the analytically extended Riemann zeta function can be expressed as:
Here we use symbols from historical documents. The integral in the formula is actually a contour integral around the positive real axis (that is, from +∞, from above the real axis to near the origin, from around the origin to below the real axis, and then from below the real axis to +∞, the distance to the real axis and the radius around the origin tend to zero). According to modern mathematical notation, it should be written as:
Where the integration path c is the same as above, and revolves around the positive real axis, which can be vividly expressed as follows:
The γ function γ (s) in the formula is a generalization of the factorial function on the complex plane. For a positive integer S >；; 1:γ(s)=(s- 1)！ . It can be proved that this integral expression is analytical in the whole complex plane except for a simple pole at s= 1. This is the complete definition of Riemann zeta function.
Using the above integral expression, it can be proved that the Riemann zeta function satisfies the following algebraic relationship:
From this relation, it is not difficult to find that the value of Riemann zeta function is zero at s=-2n (n is a positive integer)-because sin(πs/2) is zero. The point on the complex plane that makes the value of Riemann zeta function zero is called the zero point of Riemann zeta function. So s=-2n (n is a positive integer) is the zero point of the Riemann zeta function. These zeros are ordered in distribution and simple in nature, so they are called ordinary zeros of Riemann zeta function. Besides these trivial zeros, there are many other zeros in Riemannian zeta function, and their properties are far more complicated than trivial zeros, which are called nontrivial zeros.
Riemann conjecture puts forward:
All nontrivial zeros of Riemannian zeta function lie on the straight line of Re(s)= 1/2 on the complex plane. That is, the real part of the solution of equation zeta (s) = 0 is 1/2.
In the study of Riemann conjecture, mathematicians call the straight line Re(s)= 1/2 on the complex plane the critical line. Using this term, Riemann conjecture can also be expressed as: all nontrivial zeros of Riemann zeta function are located on the critical line.
Research process conjecture verification Riemann conjecture was put forward by German mathematician Bernard in 1859, which involves the distribution of prime numbers and is considered as one of the most difficult mathematical problems in the world. Three Dutch mathematicians, J.van de Lune, H. J. Rielette and D.T.Winter, used computers to test Riemann's hypothesis. They tested the zeros of the first 200 million homographs and proved that Riemann's hypothesis was correct. They published their results at 198 1, and they continued to test some zeros below them with computers.
The Mystery of the Century 1982 1 1 Soviet mathematician Marty Yexueweiqi announced in the Soviet magazine "KiBerrica" that he had tested a mathematical problem related to Riemann conjecture with a computer, which could prove that the problem was correct, thus in turn supporting that Riemann conjecture was probably correct.
Levinson of MIT 1975 proved that No (t) >: 0.3474N(T).
1980, China mathematician Lou He made a little improvement on levinson's work. They proved that NO (t) >: 0.35N(T).
In the article published by C.L.Siegel in 1932, there is the following formula:
According to the geometric meaning of this formula and the zero property of cos function, the author directly deduces No(T)=N(T), and proves that all zeros in the region fall on the critical line.
C. L. Siegel sorted out four formulas from Riemann's manuscripts, three of which often appeared in literature and textbooks, but the above formulas were rarely mentioned in the literature for more than 80 years, and even C. L. Siegel himself was puzzled by the function of this formula. In fact, as long as we jump out of analytic number theory and look at Riemann's manuscript, we can clearly see that Riemann strictly proved the modern "Riemann conjecture" with the geometric thought of complex analysis. This may be the biggest injustice in the history of mathematics.
On the 20th 1165438+1October17th, Professor Opeyemi Enoch of Nigeria successfully solved the existing mathematical problem 156- Riemann conjecture, and obtained1ten thousand dollars (about RMB 6.3 million).
In 2000, Clay Institute of Mathematics listed Riemann conjecture as one of the seven Millennium mathematical problems.
2065438+In September 2008, Michael Atia announced that he would prove the Riemann Hypothesis, which will be presented at the Heidelberg Prize Winners Forum on September 24th. Michael Atia posted a preprint of his Riemann hypothesis (conjecture) proof.
Research Achievements 2065438+On September 24th, 2008, in Heidelberg, Germany, the famous mathematician Atia said in his speech that he had proved the Riemann conjecture.
During the speech, Atia released the above picture and proved that all zeros are on the critical line by todd function reduction to absurdity. He published this research paper, which consists of five pages. In this paper, with the help of dimensionless constant α (fine structure constant) in quantum mechanics, Atia claims to have solved the Riemann conjecture in complex domain.
Atia said that he wanted to know the dimensionless constant in quantum mechanics-the fine structure constant. Because the fine structure constant is approximately equal to1137, it describes the intensity of electromagnetic interaction. For example, in hydrogen atoms, we can roughly say that the speed of electrons around the nucleus is1137 times the speed of light.
Atia pointed out that understanding the fine structure constant is only the initial motivation. The mathematical method developed in this process can understand Riemann conjecture.
Finally, at the end of the paper, Atia said that the fine structure constant and Riemann conjecture have been solved by his method. Of course, he only solved the Riemann conjecture in complex number field and Riemann conjecture in rational number field, and he still needs to study it. In addition, with the solution of Riemann conjecture, Atia thinks that bsd conjecture is also expected to be solved. Now, of course, Atia thinks that the gravitational constant G is a more difficult constant to understand.
In Riemann conjecture, we see that the real parts of nontrivial zeros are all equal to 1/2, which is a surprising constant. Although we can see why 1/2 appears from a simple symmetry relation.
1-s=s, so s= 1/2.
Gee Friedrich Bernhard (1826- 1866, German mathematician) is the founder of Riemann geometry. During his PhD, he studied complex variable functions. He extended the usual concept of function to multi-valued function and introduced the intuitive concept of multi-leaf Riemannian surface. His doctoral thesis was praised by Gauss, and it was also the basis of his work in the next ten years, including the application of complex variable function in Abel integral and θ function, the trigonometric series representation of function, the basis of differential geometry and so on.
Riemann conjecture was put forward by Riemann in 1859. In the process of proving the prime number theorem, Riemann put forward a conclusion: the zeros of Zeta function are all on the straight line Res(s) = 1/2. He gave up after his proof failed, because it had little effect on his proof of the prime number theorem. But this problem has not been solved so far, and even a simpler guess than this assumption has not been proved. Many problems in function theory and analytic number theory depend on Riemann hypothesis. The generalized Riemann hypothesis in algebraic number theory has far-reaching influence. If we can prove the Riemann hypothesis, we can solve many problems.