Chapter 143 The Most Powerful Genius in the History of Mathematics
"Using the regularity of the boundary points of the Dirichlet function to construct a function domain with regularity boundaries, and then introducing curve equations by expanding the domain to restrict the concept of dual reduction groups"
In the auditorium of Wenjin International Hotel, Artur Avila's eyes suddenly brightened after muttering a few words to himself, and he looked at Xu Chuan excitedly.
"Xu, you are indeed known as the strongest genius in the history of mathematics. You are so powerful. Using this method, you might be able to constrain and determine the functor properties of some automorphic groups."
Xu Chuan looked embarrassed. What the hell is going on with this 'most powerful genius in the history of mathematics'? Who gave him this name?
However, during the exchange and discussion, he didn't pay much attention to this. He nodded and continued following Professor Artur Avila's words:
"Moreover, the first verified instance of Langlands's functority conjecture is the functority between the automorphic representation of GL2 on the algebraic number field and the representation of the multiplicative subgroup of quaternion algebra."
"The functority proved in this classic work also proposed the relationship between the original form of Artin's conjecture and the functority conjecture. Artin's conjecture has also been reformulated as a two-dimensional complex representation of the Galois group and the GL2 automorphic group The functority conjecture between representations.”
"Therefore, Arting's conjecture points out that the Arting's L function constructed on the Gavarro group is holomorphic, and Langlands conjecture that these Arting's L functions should essentially be L functions represented by the automorphic group."
After hearing this, Professor Artur Avila fell into deep thought, but after a while, he suddenly woke up and said with half doubt and half certainty:
"If Artin's conjecture can be proved, then it will be a big step forward for Artin's L function in terms of Langlands conjecture?"
Xu Chuan nodded and said: "From the current theoretical point of view, this is indeed true."
Then, he shook his head again and said, "But."
"But it is too difficult to solve Artin's conjecture." Professor Artur Avila sighed and completed what Xu Chuan had not finished.
Xu Chuan acquiesced and said no more.
Arting's conjecture, also known as the new Mersenne conjecture, is a generalized derivative of the famous Mersenne conjecture, which is a conjecture about prime numbers.
If you have not heard of Artin's conjecture and Mason's conjecture, then most people should have heard of the familiar Goldbach's conjecture.
They are all conjectures of the same type, which can be said to be derived from prime numbers.
In mathematics, the earliest people come into contact with are natural numbers such as 0, 1, 2, 3, and 4.
Among such natural numbers, if a number is greater than 1 and cannot be divided by other natural numbers (except 0), then this number is called a prime number, also called a prime number.
Numbers that are greater than 1 but are not prime numbers are called composite numbers. 1 and 0 are special and are neither prime numbers nor composite numbers.
As early as 2,500 years ago, people at that time noticed this strange phenomenon. Euclid, the ancient Greek mathematician and the father of geometry, proposed a very classic idea in his most famous work "Elements of Geometry". prove.
That is: Euclid proved that there are infinitely many prime numbers, and proposed that a small number of prime numbers can be written in the form of "2^p-1", where the exponent p is also a prime number.
This proof is called ‘Euclidean Prime Number Theorem’ and is one of the most basic classical propositions in number theory.
Classics never go out of style. When subsequent mathematicians studied the ‘Euclidean Prime Number Theorem’, they derived various conjectures about prime numbers.
Starting from Mersenne's prime number conjecture, to Zhou's conjecture, twin prime number conjecture, Ulam's spiral, Gilbreth's conjecture to the finally extremely famous Goldbach's conjecture and so on.
There are many conjectures derived from prime numbers, but most of them have not been proven.
The new Mersenne prime conjecture that Xu Chuan and Professor Artur Avila talked about is a conjecture derived from prime numbers, also called Artin's conjecture. It is an upgraded version of the original Mersenne prime conjecture.
Among the many prime number conjectures, the difficulty is the same as the twin prime number conjecture, second only to the famous 'Goldbach's conjecture'.
[New Mersenne Prime Conjecture: For any odd natural number p, if two of the following statements are true, the remaining one will be true:
1. p=(2^k)±1 or p=(4^k)±3
2. (2^p)-1 is a prime number (Mersenne prime number)
3. [(2^p)+ 1]/ 3 is a prime number (Wagstaff prime number)]
The New Mersenne Prime Conjecture has three problems, and the three problems are closely related. If two of them can be proved, then the remaining one will naturally be true.
In the history of scientific development, the search for Mersenne prime numbers has been an important indicator of human intelligence development in the era of hand calculations.
Just like today's IQ test questions, the more Mersenne primes that can be calculated, the smarter the person is.
Because although the Mersenne prime number seems simple, when the exponent P value is large, its exploration requires not only advanced theory and skillful skills, but also arduous calculations.
The most famous one, Euler, known as the "God of Mathematics", proved that 2^31-1 is the 8th Mersenne prime number by mental arithmetic while blind;
This 10-digit prime number (ie 2147483647) was the largest known prime number in the world at that time.
It is good enough for ordinary people to be able to add, subtract, multiply and divide three-digit numbers, but Euler can push numbers to the billion level in his mind. This terrifying computing ability, brain reaction ability and problem-solving skills can be said to be worthy of the reputation of "the chosen one".
In addition, in 2013, a research team led by Curtis Cooper, a mathematician at the University of Central Missouri, discovered the largest Mersenne prime to date - 2^57885161-1 (2 to the power of 57885161 minus 1) by participating in a project called "The Great Internet Mersenne Prime Search" (GIMPS).
This prime number is also the largest prime number known so far, with 17425170 digits, 4457081 more digits than the Mersenne primes discovered before.
If it is printed out in ordinary 18-point standard font, its length can exceed 65 kilometers.
Although this number is very large, it is very small in mathematics.
Because "number" is infinite, and numbers have the concept of infinity, in mathematics, no one knows how many prime numbers there are after 2^57885161-1 (2 to the power of 57885161 minus 1).
This has lasted for a thousand years and is the most ambitious exploration in the history of mathematics: how many Mersenne primes are there and whether they are infinite. As of now, no one has been able to give an answer.
Proving the new Mersenne prime conjecture is no less difficult than the Weyl-Berry conjecture that Xu Chuan has proved before.
So far, the most difficult proof of prime number conjecture in the mathematical community is only the weak Goldbach conjecture.
That is: [Any odd number greater than 7 can be expressed as the sum of three odd prime numbers. ]
In May 2013, Harold Heoffgot, a researcher at the École Normale Supérieure in Paris, published two papers announcing the complete proof of the weak Goldbach conjecture.
In addition, in the same year, Chinese mathematician Professor Zhang Yitang also made considerable progress in proving the prime number conjecture.
His paper "Bounded Distances Between Primes" was published in the Annals of Mathematics, which solved the problem that had troubled the mathematical community for a century and a half and proved the weakened situation of the twin prime conjecture.
That is: it was found that there are infinitely many pairs of prime numbers with a difference less than 70 million.
This is the first time that someone has proved that there are infinitely many pairs of prime numbers with a distance less than a fixed value.
But for the mathematical community, both the weak Goldbach conjecture and the weak twin prime theorem are just the prelude to climbing the peak.
They are like a loud national anthem for a climber climbing Mount Everest before setting off. They can give the climber courage to a certain extent, but it is not realistic to expect to climb Mount Everest and stand on the top of the peak.
"Xu, will you try to develop in the direction of number theory?"
After a slight silence, Professor Artur Avila looked up at Xu Chuan.
If this youngest genius in the history of mathematics develops in the direction of number theory, he may have the opportunity to pick a huge fruit in the field of prime numbers?
He dare not say for sure, after all, who can be sure of such things?
Artur Avila really wants to see the day when the Goldbach conjecture is confirmed, but he doesn't want this rising star in mathematics to plunge into it for years or even decades without making any achievements.
Prime numbers have been developed for thousands of years, and countless mathematicians have rushed into this huge pit one after another, although they have proved many conjectures and solved many problems.
But from beginning to end, the most difficult problems have never been solved.
Even, there is no hope of solving them.
But if Xu Chuan continues to study spectral theory, functional analysis, and Dirichlet function, he dare not say that he will definitely make a greater contribution than the Weyl-Berry conjecture, but he will definitely be able to further expand the boundaries and expand the scope of mathematics in these fields.
But if he switches to number theory, it is uncertain.
Not every genius is like Terence Tao. At present, Xu Chuan's mathematical talent is indeed higher than Terence Tao, but no one knows what will happen after crossing fields.
Xu Chuan did not give Avila a definite answer. In the past year, he did read a lot of books related to number theory, but number theory was not in his subsequent study and research arrangements.
He prefers functions and analysis that can be applied in practice and solve physical problems, while number theory mainly studies the properties of integers, which is pure mathematics.
Of course, with the development of mathematics to this day, it is impossible to say that any mathematical field is pure mathematics, and it can always be linked to other fields.
For example, in statistical mechanics, the partition function is the basic mathematical object of study; and in the analytical theory of prime number distribution, the zeta function is the basic object.
Therefore, this unorthodox interpretation of the zeta function as a partition function points out the possible fundamental connection between prime number distribution and this branch of physics.
However, at present, the application of number theory to the field of physics is still relatively vacant, far less extensive than mathematical analysis, function transformation, and mathematical models.
Therefore, Xu Chuan is not very inclined to invest a lot of energy and time in the field of pure number theory.
But it is certain that he will study and learn about number theory.
Because number theory is not just pure number theory, there are also various branches such as analytical number theory, algebraic number theory, geometric number theory, computational number theory, arithmetic algebraic geometry, etc.
These branches are all extended from pure number theory, that is, elementary number theory combined with other mathematics.
For example, analytic number theory is the study of number theory about integer problems with the help of calculus and complex analysis (i.e. complex variable functions).
The things he talked about with Professor Avila tonight are related to analytic number theory,
because in addition to the circle method, sieve method, etc., analytic number theory also includes modular form theory related to elliptic curves, etc. Later, it developed into automorphic form theory, which was connected with representation theory.
Therefore, having a certain foundation in number theory is still very helpful for other mathematical learning.